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H.\ Sch\"onemann; Humboldt-Universit\"at
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$3 \times 3$
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$0$
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$Q$
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$a$
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$Z/32003[x,y,z]$
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$x^3,$
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$K[x,y,z]/\hbox{\rm jacob}(f)$.
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$12-4=8$.
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$1 \times 1$-minors
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$1 \times 1$-minors,
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$MD$
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$r^3/MD$.
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$M = r^3/U$,
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$M$.
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$\hbox{ann}(M) = \{a \mid aM = 0 \}$
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$\{ a \mid ar^3 \in U \}$.
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$U \colon r^3 $.
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$n$
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$n=0$
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$R^5$
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$(z^3,0,-y+4z,x+2z,0)$ and $(-xyz-y^2z-4xz^2+16z^3,-y^2,48z,48z,x+y-z)$.
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finite fields $Z/p$, $p$ a prime $\le 2147483647$,
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finite fields $\hbox{GF}(p^n)$ with $p^n$ elements, $p$ a prime, $p^n \le 2^{16}$,
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$Z/p$
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$Z/m$
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$m\in Z$
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$Z$
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$Z/ma$
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$Q[a,b,c,d]$
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$Z/7[x,y,z]$
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$Z/7[x_1,\ldots,x_6]$
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$x_1,x_2,x_3$
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$x_4,x_5,x_6$:
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$(Q[a,b,c])[x,y,z]$
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$(x,y,z)$
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$Q[x,y,z]$
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$x$
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$y$
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$z$
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$K[x,y,z]$
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$K=Z/7(a,b,c)$
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$Z/7$
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$b$
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$c$
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$K=Z/7[a]$
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$a.$
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$K$
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$\mu_a=a^2+a+3$,
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$R[x,y,z]$
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$R$
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$R(j)[x,y,z]$
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$j$
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$R(i)[x,y,z]$
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$i$
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$Z[x,y,z]$
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$Z/6^3[x,y,z]$
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$Z/100[x,y,z]$
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$\hbox{GF}(p^n)$ with $p^n$ elements, where $p^n$ has to be less than or equal to $2^{15}$.
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$\hbox{GF}(p^n)$
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$K[x_1,\ldots,x_n]$,
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$\hbox{Loc}_{(x)}K[x_1,\ldots,x_n])$.
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$1 < x$
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$Z/n$
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$Z_p$
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$Z/p^n$
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$ m \times n $
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\quad
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$i+i+i$
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$\alpha, \beta$
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$i_1$
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$i_{1,1}$
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$\alpha$
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$ i_{1,1} $
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$(h1+h2)/h1 \cong h2/(h1 \cap h2)$
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$Q \rightarrow  Q(a, \ldots)$
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$Q \rightarrow R$
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$Q \rightarrow  C$
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$C$
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$Z/p \rightarrow  (Z/p)(a, \ldots)$
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$Z/p \rightarrow  GF(p^n)$
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$GF$
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$Z/p \rightarrow  R$
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$R \rightarrow C$
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$Z/p \rightarrow Q,
\quad
[i]_p \mapsto i \in [-p/2, \, p/2]
\subseteq Z$
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$Z/p \rightarrow Z/p^\prime,
\quad
[i]_p \mapsto i \in [-p/2, \, p/2] \subseteq Z, \;
i \mapsto [i]_{p^\prime} \in Z/p^\prime$
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$C \rightarrow R, \quad$ by taking the real part
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$Q \rightarrow Z/p$
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$Q \rightarrow (Z/p)(a, \ldots)$
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$f \circ g$
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$M$
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$R^n$
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$v_1, \ldots, v_k$
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$R^n/M$
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$\times$
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$(h1+h2)/h1=h2/(h1 \cap h2)$
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$k[X,Y]$
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$k_1[X]$
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$k_2[Y]$
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$k_1$
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$k_2$
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$R^n/M$, if $R$ denotes the basering,
$M$ is a homogeneous submodule of $R^n$ and the argument represents a
resolution of
$R^n/M$.
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The entry d of the intmat at place (i,j) is the minimal number of
generators in degree i+j of the j-th syzygy module (= module of
relations) of $R^n/M$, i.e. the 0th (resp.\ 1st) syzygy module of $R^n/M$ is
$R^n$ (resp.\ $M$).
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$$
0 \longleftarrow r/j \longleftarrow r(1)
\buildrel{T[1]}\over{\longleftarrow} r(2) \oplus r^3(3)
\buildrel{T[2]}\over{\longleftarrow} r^4(4)
\buildrel{T[3]}\over{\longleftarrow} r(5)
\longleftarrow 0 \quad .
$$
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  $2\times n$
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  $f = M[1,1]\cdot M[2,1]+\ldots+M[1,n]\cdot M[2,n].$
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  $(m_{ij})$
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\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 107 -->
\end{rawhtml}
  $(d\times 1)$-matrix and $m_{i+1,j}, 0\leq i\leq e$,
\begin{rawhtml}
<!-- l2h_end sing_l2h 107 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 108 -->
\end{rawhtml}
  $z^i$
\begin{rawhtml}
<!-- l2h_end sing_l2h 108 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 109 -->
\end{rawhtml}
  $f_1, f_2,\ldots,f_k$
\begin{rawhtml}
<!-- l2h_end sing_l2h 109 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 110 -->
\end{rawhtml}
  $(d\times k)$-matrix and $m_{i+1,j}, 0\leq i\leq e, 1\leq j\leq k$,
\begin{rawhtml}
<!-- l2h_end sing_l2h 110 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 111 -->
\end{rawhtml}
  $f_j$.
\begin{rawhtml}
<!-- l2h_end sing_l2h 111 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 112 -->
\end{rawhtml}
  $(dk\times 1)$-matrix and $m_{(j-1)d+i+1,1}, 0\leq i\leq e, 1\leq j\leq k$,
\begin{rawhtml}
<!-- l2h_end sing_l2h 112 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 113 -->
\end{rawhtml}
  $v_1, v_2,\ldots,v_s$
\begin{rawhtml}
<!-- l2h_end sing_l2h 113 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 114 -->
\end{rawhtml}
  $(dk\times s)$-matrix and $m_{(j-1)d+i+1,r}, 0\leq i\leq e, 1\leq j\leq k, 1\leq r\leq s$,
\begin{rawhtml}
<!-- l2h_end sing_l2h 114 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 115 -->
\end{rawhtml}
  $v_r$.
\begin{rawhtml}
<!-- l2h_end sing_l2h 115 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 116 -->
\end{rawhtml}
  $\sum_{i=1}^k m_i\cdot a_i\cdot x^{e_i}$,
\begin{rawhtml}
<!-- l2h_end sing_l2h 116 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 117 -->
\end{rawhtml}
  $m_i$
\begin{rawhtml}
<!-- l2h_end sing_l2h 117 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 118 -->
\end{rawhtml}
  $a_i$
\begin{rawhtml}
<!-- l2h_end sing_l2h 118 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 119 -->
\end{rawhtml}
  $x^{e_i}$
\begin{rawhtml}
<!-- l2h_end sing_l2h 119 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 120 -->
\end{rawhtml}
  $\sum_{i=1}^k m_i\cdot a_i$,
\begin{rawhtml}
<!-- l2h_end sing_l2h 120 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 121 -->
\end{rawhtml}
  $K\cdot A=M$.
\begin{rawhtml}
<!-- l2h_end sing_l2h 121 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 122 -->
\end{rawhtml}
  $K\cdot A$
\begin{rawhtml}
<!-- l2h_end sing_l2h 122 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 123 -->
\end{rawhtml}
  $m\times n$
\begin{rawhtml}
<!-- l2h_end sing_l2h 123 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 124 -->
\end{rawhtml}
$${\rm contract}(x^A ,  x^B) := \cases{ x^{(B-A)}, &if $B\ge A$
componentwise\cr 0,&otherwise.\cr}$$
\begin{rawhtml}
<!-- l2h_end sing_l2h 124 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 125 -->
\end{rawhtml}
$x$.
\begin{rawhtml}
<!-- l2h_end sing_l2h 125 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 126 -->
\end{rawhtml}
$|a|,|b|<=\sqrt{(N-1)/2}$
\begin{rawhtml}
<!-- l2h_end sing_l2h 126 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 127 -->
\end{rawhtml}
$x_i$
\begin{rawhtml}
<!-- l2h_end sing_l2h 127 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 128 -->
\end{rawhtml}
$x_i>1$ then $x_i$
\begin{rawhtml}
<!-- l2h_end sing_l2h 128 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 129 -->
\end{rawhtml}
$f_1,\dots,f_k$ of I, let $f'_i$ be obtained from
$f_i$ by deleting the terms divisible by $x_i\cdot m$ for all i with $x_i<1$.
Then $f'_1,\dots,f'_k$ generate I.
\begin{rawhtml}
<!-- l2h_end sing_l2h 129 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 130 -->
\end{rawhtml}
$$...\longrightarrow F_2 \buildrel{A_2}\over{\longrightarrow} F_1
\buildrel{A_1}\over{\longrightarrow} R\longrightarrow R/I
\longrightarrow 0.$$
\begin{rawhtml}
<!-- l2h_end sing_l2h 130 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 131 -->
\end{rawhtml}
$M_i={\tt module} (A_i)$, i=1..k.
\begin{rawhtml}
<!-- l2h_end sing_l2h 131 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 132 -->
\end{rawhtml}
${\tt L[i]}=M_i$
\begin{rawhtml}
<!-- l2h_end sing_l2h 132 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 133 -->
\end{rawhtml}
$I \cap K[U]=(0)$,
\begin{rawhtml}
<!-- l2h_end sing_l2h 133 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 134 -->
\end{rawhtml}
Input: $f_1,\dots,f_n$
\begin{rawhtml}
<!-- l2h_end sing_l2h 134 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 135 -->
\end{rawhtml}
Output: $g_1,\dots,g_s$ with $s \leq n$ and the properties
\begin{rawhtml}
<!-- l2h_end sing_l2h 135 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 136 -->
\end{rawhtml}
$(f_1,\dots,f_n) = (g_1,\dots,g_s)$,
\begin{rawhtml}
<!-- l2h_end sing_l2h 136 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 137 -->
\end{rawhtml}
$L(g_i)\neq L(g_j)$ for all $i\neq j$,
\begin{rawhtml}
<!-- l2h_end sing_l2h 137 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 138 -->
\end{rawhtml}
$L(g_i)$
\begin{rawhtml}
<!-- l2h_end sing_l2h 138 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 139 -->
\end{rawhtml}
$\{g_1,\dots,g_{i-1},g_{i+1},\dots,g_s\}$,
\begin{rawhtml}
<!-- l2h_end sing_l2h 139 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 140 -->
\end{rawhtml}
$L(g_i) | L(g_j)$ for any $i \neq j$,
\begin{rawhtml}
<!-- l2h_end sing_l2h 140 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 141 -->
\end{rawhtml}
$ecart(g_i) > ecart(g_j)$.
\begin{rawhtml}
<!-- l2h_end sing_l2h 141 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 142 -->
\end{rawhtml}
Here, $L(g)$ denotes the leading term of $g$ and
$ecart(g):=deg(g)-deg(L(g))$.
\begin{rawhtml}
<!-- l2h_end sing_l2h 142 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 143 -->
\end{rawhtml}
$M_i={\tt module}(A_i)$, i=1..k.
\begin{rawhtml}
<!-- l2h_end sing_l2h 143 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 144 -->
\end{rawhtml}
$K[x_1,x_2,\ldots,x_r]$,
\begin{rawhtml}
<!-- l2h_end sing_l2h 144 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 145 -->
\end{rawhtml}
$A^{-1}=U^{-1}\cdot L^{-1}\cdot P$,
\begin{rawhtml}
<!-- l2h_end sing_l2h 145 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 146 -->
\end{rawhtml}
$1, A^{-1}$.
\begin{rawhtml}
<!-- l2h_end sing_l2h 146 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 147 -->
\end{rawhtml}
represents $h_1/(h_1 \cap h_2) \cong (h_1+h_2)/h_2$
\begin{rawhtml}
<!-- l2h_end sing_l2h 147 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 148 -->
\end{rawhtml}
$h_1$ and $h_2$
\begin{rawhtml}
<!-- l2h_end sing_l2h 148 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 149 -->
\end{rawhtml}
$R^l$
\begin{rawhtml}
<!-- l2h_end sing_l2h 149 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 150 -->
\end{rawhtml}
$H_1$, resp.\ $H_2$,
\begin{rawhtml}
<!-- l2h_end sing_l2h 150 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 151 -->
\end{rawhtml}
be the matrices of size $l \times k$, resp.\ $l \times m$, having the
generators of $h_1$, resp.\ $h_2$,
\begin{rawhtml}
<!-- l2h_end sing_l2h 151 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 152 -->
\end{rawhtml}
$h_1/(h_1 \cap h_2) \cong R^k / ker(\overline{H_1})$
\begin{rawhtml}
<!-- l2h_end sing_l2h 152 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 153 -->
\end{rawhtml}
$\overline{H_1}: R^k \rightarrow R^l/Im(H_2)=R^l/h_2$
is the induced map.
\begin{rawhtml}
<!-- l2h_end sing_l2h 153 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 154 -->
\end{rawhtml}
$coker(A)=F_0/M$
$$...\longrightarrow F_2 \buildrel{A_2}\over{\longrightarrow} F_1
\buildrel{A_1}\over{\longrightarrow} F_0\longrightarrow F_0/M
\longrightarrow 0,$$
\begin{rawhtml}
<!-- l2h_end sing_l2h 154 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 155 -->
\end{rawhtml}
$A_1$
\begin{rawhtml}
<!-- l2h_end sing_l2h 155 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 156 -->
\end{rawhtml}
$M_i={\tt module}(A_i)$, i=1...k.
\begin{rawhtml}
<!-- l2h_end sing_l2h 156 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 157 -->
\end{rawhtml}
${\tt L[i]}\neq 0$ for $i \le p$,
\begin{rawhtml}
<!-- l2h_end sing_l2h 157 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 158 -->
\end{rawhtml}
$A_1$=matrix(M),
\begin{rawhtml}
<!-- l2h_end sing_l2h 158 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 159 -->
\end{rawhtml}
$coker(A_1)=F_0/M$
$$...\longrightarrow F_2 \buildrel{A_2}\over{\longrightarrow} F_1 \buildrel{A_1}\over{\longrightarrow} F_0\longrightarrow F_0/M\longrightarrow 0,$$
\begin{rawhtml}
<!-- l2h_end sing_l2h 159 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 160 -->
\end{rawhtml}
$M_i={\tt module}(A_i),\;i = 1,\ldots,k$.
\begin{rawhtml}
<!-- l2h_end sing_l2h 160 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 161 -->
\end{rawhtml}
(${\tt L[i]}=M_i$
\begin{rawhtml}
<!-- l2h_end sing_l2h 161 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 162 -->
\end{rawhtml}
$d$ in row $i$ and column $j$
\begin{rawhtml}
<!-- l2h_end sing_l2h 162 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 163 -->
\end{rawhtml}
$i+j$ of the $j$-th
\begin{rawhtml}
<!-- l2h_end sing_l2h 163 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 164 -->
\end{rawhtml}
$R^n/M$ (the 0th and 1st syzygy module of $R^n/M$ is $R^n$ and $M$, resp.).
\begin{rawhtml}
<!-- l2h_end sing_l2h 164 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 165 -->
\end{rawhtml}
${\tt R}^n$.
\begin{rawhtml}
<!-- l2h_end sing_l2h 165 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 166 -->
\end{rawhtml}
$\{a \in R \mid aJ \subset I\}$,
\begin{rawhtml}
<!-- l2h_end sing_l2h 166 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 167 -->
\end{rawhtml}
$\{b \in R \mid bN \subset M\}$.
\begin{rawhtml}
<!-- l2h_end sing_l2h 167 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 168 -->
\end{rawhtml}
$\{b \in R^n \mid Jn \subset M\}$.
\begin{rawhtml}
<!-- l2h_end sing_l2h 168 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 169 -->
\end{rawhtml}
\noindent
Let $0 \rightarrow\ \bigoplus_a K[x]e_{a,n}\ \rightarrow\ \dots
  \rightarrow\ \bigoplus_a K[x]e_{a,0}\ \rightarrow\
  I\ \rightarrow\ 0$
be a minimal resolution of I considered with homogeneous maps of degree 0.
The regularity is the smallest number $s$ with the property deg($e_{a,i})
 \leq s+i$ for all $i$.
\begin{rawhtml}
<!-- l2h_end sing_l2h 169 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 170 -->
\end{rawhtml}
computes the permutation {\tt v}
which orders the ideal, resp.\ module, {\tt I} by its initial terms,
starting with the smallest, that is, {\tt I(v[i]) < I(v[i+1])} for all
{\tt i}.
\begin{rawhtml}
<!-- l2h_end sing_l2h 170 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 171 -->
\end{rawhtml}
$A_1={\tt matrix}(M)$.
\begin{rawhtml}
<!-- l2h_end sing_l2h 171 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 172 -->
\end{rawhtml}
$coker(A_1)=F_0/M$
$$...\longrightarrow F_2 \buildrel{A_2}\over{\longrightarrow} F_1 \buildrel{A_1}\over{\longrightarrow} F_0\longrightarrow F_0/M\longrightarrow 0.$$
\begin{rawhtml}
<!-- l2h_end sing_l2h 172 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 173 -->
\end{rawhtml}
{\tt vandermonde(p,v,d)} computes the (unique) polynomial of degree
{\tt d} with prescribed values {\tt v[1],...,v[N]} at the points
{\tt p}$^0,\dots,$ {\tt p}$^{N-1}$, {\tt N=(d+1)}$^n$, $n$ the
number of ring variables.

The returned polynomial is $\sum
c_{\alpha_1\ldots\alpha_n}\cdot x_1^{\alpha_1} \cdot \dots \cdot
x_n^{\alpha_n}$, where the coefficients
$c_{\alpha_1\ldots\alpha_n}$ are the solution of the (transposed)
Vandermonde system of linear equations
$$ \sum_{\alpha_1+\ldots+\alpha_n\leq d} c_{\alpha_1\ldots\alpha_n} \cdot
{\tt p}_1^{(k-1)\alpha_1}\cdot\dots\cdot {\tt p}_n^{(k-1)\alpha_n} =
{\tt v}[k], \quad k=1,\dots,{\tt N}.$$
\begin{rawhtml}
<!-- l2h_end sing_l2h 173 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 174 -->
\end{rawhtml}
the ground field has to be the field of rational
numbers. Moreover, {\tt ncols(p)==}$n$, the number of variables in the
basering, and all the given generators have to be numbers different from
0,1 or -1. Finally, {\tt ncols(v)==(d+1)$^n$}, and all given generators have
to be numbers.
\begin{rawhtml}
<!-- l2h_end sing_l2h 174 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 175 -->
\end{rawhtml}
$G-$
\begin{rawhtml}
<!-- l2h_end sing_l2h 175 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 176 -->
\end{rawhtml}
$GR-$
\begin{rawhtml}
<!-- l2h_end sing_l2h 176 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 177 -->
\end{rawhtml}
$G$
\begin{rawhtml}
<!-- l2h_end sing_l2h 177 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 178 -->
\end{rawhtml}
$A$
\begin{rawhtml}
<!-- l2h_end sing_l2h 178 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 179 -->
\end{rawhtml}
$GR$
\begin{rawhtml}
<!-- l2h_end sing_l2h 179 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 180 -->
\end{rawhtml}
$A^n$.
\begin{rawhtml}
<!-- l2h_end sing_l2h 180 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 181 -->
\end{rawhtml}
$Z/p \rightarrow Q, \quad [i]_p \mapsto i \in [-p/2, \, p/2] \subseteq Z$
\begin{rawhtml}
<!-- l2h_end sing_l2h 181 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 182 -->
\end{rawhtml}
$v_1, \ldots, v_k$, then
\begin{rawhtml}
<!-- l2h_end sing_l2h 182 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 183 -->
\end{rawhtml}
$n \times k$,
\begin{rawhtml}
<!-- l2h_end sing_l2h 183 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 184 -->
\end{rawhtml}
$v_1, \ldots, v_k$ which generate $M$.
\begin{rawhtml}
<!-- l2h_end sing_l2h 184 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 185 -->
\end{rawhtml}
$C = A\otimes_{\bf{K} } B$
\begin{rawhtml}
<!-- l2h_end sing_l2h 185 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 186 -->
\end{rawhtml}
$G$-algebras $A$ and $B$
\begin{rawhtml}
<!-- l2h_end sing_l2h 186 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 187 -->
\end{rawhtml}
$A= k_1 \langle x_1, \ldots ,x_n \mid$
$\{ x_j x_i=c_{ij} \cdot x_i x_j + d_{ij}\}, 1 \leq i <j \leq n \rangle$,
and
$B= k_2 \langle y_1, \ldots ,y_m \mid$
$\{ y_j y_i=q_{ij} \cdot y_i y_j + r_{ij}\}, 1 \leq i <j \leq m \rangle$
\begin{rawhtml}
<!-- l2h_end sing_l2h 187 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 188 -->
\end{rawhtml}
$C = K \langle x_1, \ldots ,x_n, y_1, \ldots ,y_m \mid$
$\{ x_j x_i=c_{ij} \cdot x_i x_j + d_{ij}, 1 \leq i <j \leq n\}$,
$\{ y_j y_i=q_{ij} \cdot y_i y_j + r_{ij}, 1 \leq i <j \leq m\}$,
$\{ y_j x_i = x_i y_j, 1 \leq j \leq m, 1 \leq i \leq n\}  \rangle$.
\begin{rawhtml}
<!-- l2h_end sing_l2h 188 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 189 -->
\end{rawhtml}
$B$
\begin{rawhtml}
<!-- l2h_end sing_l2h 189 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 190 -->
\end{rawhtml}
$A\otimes_{\bf{K} } B$
\begin{rawhtml}
<!-- l2h_end sing_l2h 190 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 191 -->
\end{rawhtml}
$R^n/M$, if $R$ denotes the basering and
$M$ a homogeneous submodule of $R^n$ and the argument represents a
resolution of
$R^n/M$.
\begin{rawhtml}
<!-- l2h_end sing_l2h 191 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 192 -->
\end{rawhtml}
The entry d of the intmat at place (i, j) is the minimal number of
generators in degree i+j of the j-th syzygy module (= module of
relations) of $R^n/M$ (the 0th (resp.\ 1st) syzygy module of $R^n/M$ is
$R^n$ (resp.\ $M$)).
\begin{rawhtml}
<!-- l2h_end sing_l2h 192 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 193 -->
\end{rawhtml}
$A^{env} = A \otimes_K A^{opp}$, where $A^{opp}$ is the
opposite algebra of $A$.
\begin{rawhtml}
<!-- l2h_end sing_l2h 193 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 194 -->
\end{rawhtml}
$H_1$ (resp.\ $H_2$)
\begin{rawhtml}
<!-- l2h_end sing_l2h 194 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 195 -->
\end{rawhtml}
 be the matrix of size $l \times k$ (resp.\ $l \times m$), having the
generators of $h_1$ (resp.\ $h_2$)
\begin{rawhtml}
<!-- l2h_end sing_l2h 195 -->
\end{rawhtml}
\begin{rawhtml}
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$\overline{H_1}: R^k \rightarrow R^l/Im(H_2)=R^l/h_2$
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\end{rawhtml}
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$H_1$.
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\end{rawhtml}
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$h_1$ or $h_2$,
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\end{rawhtml}
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$M_i={\tt module}(A_i)$, $i= 1 \ldots k$.
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\end{rawhtml}
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$ x_1, \ldots, x_k ,\;$
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\end{rawhtml}
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$ \{ x_j x_i=c_{ij} \cdot x_i x_j + d_{ij}, 1 \leq i <j \leq k \}, $
where $c_{ij}$ and $d_{ij}$ must be put into two strictly upper
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\end{rawhtml}
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$c_{ij}$
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\end{rawhtml}
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\end{rawhtml}
$ d_{ij}$
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\end{rawhtml}
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\end{rawhtml}
$\forall i<j$, $c_{ij} = n$,
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\end{rawhtml}
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$\forall i<j$, $d_{ij} = p$,
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\end{rawhtml}
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\end{rawhtml}
 $O, \; (O^{opp})^{opp} = O$.
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\end{rawhtml}
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\end{rawhtml}
$\forall \; f,g \in A^{opp}$, a new multiplication $*$ on $A^{opp}$
is defined to be $f*g := g \sharp f$.
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\end{rawhtml}
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<!-- l2h_begin sing_l2h 208 -->
\end{rawhtml}
$\{b \in R^n \mid bJ \subset M\}$.
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\end{rawhtml}
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\end{rawhtml}
 Let $K$ be a field, and let a $K$-algebra $A$ be generated by
variables $x_1, \ldots ,x_n$ subject to some relations.
We call $A$ an algebra with {\bf PBW basis} (Poincar\'e-Birkhoff-Witt basis), if a $K$-basis of
$A$ is Mon$(x_1,\dots,x_n)=\{x^{a_1}_1 x^{a_2}_2 \dots x^{a_n}_n \mid a_i \in N \cup \{0\} \}$,
where a power-product $x^{a_1}_1 x^{a_2}_2 \dots x^{a_n}_n$ (in this particular order) is called
{\bf a monomial}. For example, $x_1 x_2$ is a monomial, while $x_2 x_1$ is, in general, not
a monomial.
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\end{rawhtml}
 Let $K$ be a field, and let a $K$-algebra $A$ be given in terms of generators subject to the following relations:

$A= K \langle x_1, \ldots ,x_n \mid \{ x_j x_i=c_{ij} \cdot x_i x_j + d_{ij}\}, 1 \leq i <j \leq n \rangle$,
where $c_{ij} \in K^{*}, d_{ij} \in K[x_1, \ldots, x_n]$.

 $A$ is called \textbf{a $G$--algebra}, if the following conditions hold:
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there is a monomial well-ordering $<$ on $K[x_1,x_2,\ldots,x_n]$ such that
                        $\forall i<j \;\; \hbox{LM}(d_{ij})< x_i x_j$,
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$\forall \; 1 \leq i<j<k \leq n \; \; : \; {\cal NDC}_{ijk} =0$, where
$$
{\cal NDC}_{ijk} = c_{ik}c_{jk} \cdot d_{ij}x_k - x_k d_{ij} +
c_{jk} \cdot x_j d_{ik} - c_{ij} \cdot d_{ik}x_j + d_{jk}x_i
- c_{ij}c_{ik} \cdot x_i d_{jk}.
$$
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\end{rawhtml}
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\end{rawhtml}
Let $A$ be a $G$-algebra. Then
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\end{rawhtml}
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$A$ has a PBW (Poincar\'e-Birkhoff-Witt) basis,
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\end{rawhtml}
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$A$
is left and right noetherian,
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\end{rawhtml}
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\end{rawhtml}
$A$ is an integral domain.
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\end{rawhtml}
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\end{rawhtml}
In order to set up a $G$--algebra one has to do the following steps:
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\end{rawhtml}
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\end{rawhtml}
$R= K[x_1, \ldots, x_n]$,
\begin{rawhtml}
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\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 219 -->
\end{rawhtml}
$<$
\begin{rawhtml}
<!-- l2h_end sing_l2h 219 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 220 -->
\end{rawhtml}
 $\{x_i\}$
\begin{rawhtml}
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\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 221 -->
\end{rawhtml}
Poincar\'e
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\end{rawhtml}
\begin{rawhtml}
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\end{rawhtml}
$\{x^{a_1}_1 x^{a_2}_2 \dots x^{a_n}_n\}$.
\begin{rawhtml}
<!-- l2h_end sing_l2h 222 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 223 -->
\end{rawhtml}
$n\times n$
\begin{rawhtml}
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\end{rawhtml}
\begin{rawhtml}
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\end{rawhtml}
 $C=\{c_{ij}, i<j\}$, with nonzero entries $c_{ij}$ of type number ($c_{ij}$ for $i\geq j$ will be ignored).
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\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 225 -->
\end{rawhtml}
  $D=\{d_{ij}, i<j\}$, with polynomial entries $d_{ij}$ from $R$ ($d_{ij}$ for $i\geq j$ will be ignored).
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<!-- l2h_end sing_l2h 225 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 226 -->
\end{rawhtml}
$C$ and $D$.
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\end{rawhtml}
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\end{rawhtml}
For a $G$--algebra $A$, we denote by ${}_{A} \langle g_1, \dots, g_s \rangle$
the left submodule of a free module $A^r$, generated by elements $\{g_1, \dots, g_s\}\subset A^r$.
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\end{rawhtml}
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\end{rawhtml}
Let $<$ be a fixed monomial well-ordering on the $G$-algebra $A$ with the Poincar@'e-Birkhoff-Witt (PBW) basis
$\{x^{\alpha} = x^{a_1}_1 x^{a_2}_2 \dots x^{a_n}_n\}$.
For a given free module $A^r$ with the basis $\{e_1,\ldots, e_r\}$, $<$ denotes also a
fixed module ordering on the set of monomials $\{x^{\alpha} e_i \mid \alpha\in {\bf N}^n, 1\leq i\leq r \}$.
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\end{rawhtml}
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\end{rawhtml}
For a set $S \subset A^r$, define
$L(S)$ to be the $K$-vector space, spanned on the leading monomials
of elements of $S$, $L(S) = \oplus \{K x^{\alpha} e_i \mid \exists s \in S, \hbox{LM(s)}=x^{\alpha}e_i\}$.

We call $L(S)$ the \textbf{span of leading monomials} of $S$.
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\end{rawhtml}
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\end{rawhtml}
Let $I \subset A^r$ be a left $A$-submodule.
A finite set $G\subset I$ is called {\bf a left  Groebner basis} of $I$ if and
only if $L(G)=L(I)$, that is for any $f \in I\setminus \{ 0 \}$ there exists a $g\in G$
satisfying $ \hbox{LM}(g) \mid \hbox{LM}(f)$, i.e., if $\hbox{LM}(f) = x^{\alpha}e_i$, then
$\hbox{LM}(f) = x^{\beta}e_i$ with $\beta_j \leq \alpha_j, \; 1\leq j \leq n$.
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A Groebner basis $G\subset A^r$ is called {\bf minimal} (or {\bf reduced}) if $0\notin G$ and if
$\hbox{LM}(g)\notin L(G\setminus \{ g \})$ for all $g\in G$.
Note, that any Groebner basis can be made minimal by deleting successively those
$g$ with $\hbox{LM}(h)\mid \hbox{LM}(g)$ for some $h\in G\setminus\{g \}$.

For $f\in A^r $ and $ G\subset A^r $ we say that $f$ is {\bf completely reduced with
respect to $G$} if no monomial of $f$ is contained in $L(G)$.
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A map $\hbox{NF} : A^r \times \{G \mid G\ \hbox{ a (left) Groebner
basis}\} \to A^r, (f|G) \mapsto \hbox{NF}(f|G)$, is called a {\bf (left) normal form}
on $A^r$ if for any $f \in A^r$ and any left Groebner basis $G$ the following
holds:

(i) $\hbox{NF}(0|G) = 0$,

(ii)  if $\hbox{NF}(f|G) \not= 0$ then $\hbox{LM}(g)$ does not divide
$\hbox{LM}(\hbox{NF}(f|G))$ for all $g \in G$,

(iii) $f - \hbox{NF}(f|G)\in {}_{A}\langle G \rangle$.

\noindent
$\hbox{NF}(f|G)$ is called a {\bf left normal form of} $f$ {\bf with
respect to} $G$ (note that such a map is not unique).
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For a left Groebner basis $G$ of $I$ the following holds:
$f \in I$ if and only if the left normal form $\hbox{NF}(f|G) = 0$.
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\end{rawhtml}
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\end{rawhtml}
Let $J$ be a two-sided ideal and $T$ be a two-sided Groebner basis of $J$.

Then $f \in J$ if and only if the left normal form $\hbox{NF}(f|T) = 0$.
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\end{rawhtml}
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<!-- l2h_begin sing_l2h 235 -->
\end{rawhtml}
Let $A$ be a GR-algebra. A {\bf left} (resp. {\bf right}) {\bf syzygy}
between $k$ elements $\{f_1,\dots,f_k\} \subset A^r $ is
a $k$-tuple $(g_1,\dots ,g_k)\in A^k$ satisfying
$$\sum_{i=1}^{k} g_i f_i = 0 \;\;\; {\rm resp. } \;\;\; \sum_{i=1}^{k} f_i g_i = 0. $$
The set of all left (resp. right) syzygies between $\{f_1,...,f_k\}$ is a left (resp. right) submodule $S$ of $A^k$.
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\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 236 -->
\end{rawhtml}
$M \cdot S \not=0$).
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<!-- l2h_end sing_l2h 236 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 237 -->
\end{rawhtml}
$I$
\begin{rawhtml}
<!-- l2h_end sing_l2h 237 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 238 -->
\end{rawhtml}
$\{g_1, \dots , g_s\}$,
\begin{rawhtml}
<!-- l2h_end sing_l2h 238 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 239 -->
\end{rawhtml}
Let $I={}_{A}\langle g_1,\dots ,g_s\rangle \subseteq A^r$ and $M= A^r/I$.
A {\bf free resolution of $M$} is a long exact sequence
$$\dots \longrightarrow F_2 \buildrel{B_2}\over{\longrightarrow} F_1
 \buildrel{B_1}\over{\longrightarrow} F_0\longrightarrow M\longrightarrow
0,$$
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\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 240 -->
\end{rawhtml}
${\tt transpose}(B_{i+1}) \cdot {\tt transpose}(B_i) =0$
\begin{rawhtml}
<!-- l2h_end sing_l2h 240 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 241 -->
\end{rawhtml}
$B_1$
\begin{rawhtml}
<!-- l2h_end sing_l2h 241 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 242 -->
\end{rawhtml}
For a $G$-algebra $A$, generated by $n$
\begin{rawhtml}
<!-- l2h_end sing_l2h 242 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 243 -->
\end{rawhtml}
$Z/2Z$
\begin{rawhtml}
<!-- l2h_end sing_l2h 243 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 244 -->
\end{rawhtml}
$1$
\begin{rawhtml}
<!-- l2h_end sing_l2h 244 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 245 -->
\end{rawhtml}
$m$
\begin{rawhtml}
<!-- l2h_end sing_l2h 245 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 246 -->
\end{rawhtml}
$$K\left\langle x_1, \ldots, x_n; y_1, \ldots, y_m \mid y_j * y_i = - y_i y_j, i<j \right\rangle / \left\langle y_1^2, \ldots, y_m^2  \right\rangle.$$
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<!-- l2h_end sing_l2h 246 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 247 -->
\end{rawhtml}
$m > 0$
\begin{rawhtml}
<!-- l2h_end sing_l2h 247 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 248 -->
\end{rawhtml}
$y_i * y_i = 0$
\begin{rawhtml}
<!-- l2h_end sing_l2h 248 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 249 -->
\end{rawhtml}
$r$
\begin{rawhtml}
<!-- l2h_end sing_l2h 249 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 250 -->
\end{rawhtml}
$<x_1$
\begin{rawhtml}
<!-- l2h_end sing_l2h 250 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 251 -->
\end{rawhtml}
$x_n >$
\begin{rawhtml}
<!-- l2h_end sing_l2h 251 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 252 -->
\end{rawhtml}
$x_1$
\begin{rawhtml}
<!-- l2h_end sing_l2h 252 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 253 -->
\end{rawhtml}
$x_n$
\begin{rawhtml}
<!-- l2h_end sing_l2h 253 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 254 -->
\end{rawhtml}
$d$
\begin{rawhtml}
<!-- l2h_end sing_l2h 254 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 255 -->
\end{rawhtml}
$K[X]$
\begin{rawhtml}
<!-- l2h_end sing_l2h 255 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 256 -->
\end{rawhtml}
$k$
\begin{rawhtml}
<!-- l2h_end sing_l2h 256 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 257 -->
\end{rawhtml}
$K<X>$
\begin{rawhtml}
<!-- l2h_end sing_l2h 257 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 258 -->
\end{rawhtml}
$K<x,y,z>/J$
\begin{rawhtml}
<!-- l2h_end sing_l2h 258 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 259 -->
\end{rawhtml}
$Z<x,y>$
\begin{rawhtml}
<!-- l2h_end sing_l2h 259 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 260 -->
\end{rawhtml}
$<2x,3y>$
\begin{rawhtml}
<!-- l2h_end sing_l2h 260 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 261 -->
\end{rawhtml}
$Z<x,y>/<2x,3y> =$
\begin{rawhtml}
<!-- l2h_end sing_l2h 261 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 262 -->
\end{rawhtml}
$Z<x,y>/<2x,3y,yx,xy> =$
\begin{rawhtml}
<!-- l2h_end sing_l2h 262 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 263 -->
\end{rawhtml}
$Z<x,y>/<2x,3y,yx-xy,xy>$
\begin{rawhtml}
<!-- l2h_end sing_l2h 263 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 264 -->
\end{rawhtml}
$Z[x,y]/<2x,3y,xy>$
\begin{rawhtml}
<!-- l2h_end sing_l2h 264 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 265 -->
\end{rawhtml}
$y z^i x$
\begin{rawhtml}
<!-- l2h_end sing_l2h 265 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 266 -->
\end{rawhtml}
$x z^i y$
\begin{rawhtml}
<!-- l2h_end sing_l2h 266 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 267 -->
\end{rawhtml}
$i>=0$
\begin{rawhtml}
<!-- l2h_end sing_l2h 267 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 268 -->
\end{rawhtml}
$y*z^i*x$
\begin{rawhtml}
<!-- l2h_end sing_l2h 268 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 269 -->
\end{rawhtml}
$x*z^i*y$
\begin{rawhtml}
<!-- l2h_end sing_l2h 269 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 270 -->
\end{rawhtml}
$-y z^i*(2x) + (3y)*z^i x = y z^i x$
\begin{rawhtml}
<!-- l2h_end sing_l2h 270 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 271 -->
\end{rawhtml}
$Q<x,y,z>/J$
\begin{rawhtml}
<!-- l2h_end sing_l2h 271 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 272 -->
\end{rawhtml}
$i>=2$
\begin{rawhtml}
<!-- l2h_end sing_l2h 272 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 273 -->
\end{rawhtml}
$J$
\begin{rawhtml}
<!-- l2h_end sing_l2h 273 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 274 -->
\end{rawhtml}
$x y^i z$
\begin{rawhtml}
<!-- l2h_end sing_l2h 274 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 275 -->
\end{rawhtml}
$V$
\begin{rawhtml}
<!-- l2h_end sing_l2h 275 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 276 -->
\end{rawhtml}
$< x_1$
\begin{rawhtml}
<!-- l2h_end sing_l2h 276 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 277 -->
\end{rawhtml}
$T(V)$
\begin{rawhtml}
<!-- l2h_end sing_l2h 277 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 278 -->
\end{rawhtml}
$n>1$
\begin{rawhtml}
<!-- l2h_end sing_l2h 278 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 279 -->
\end{rawhtml}
$x_n > /I$
\begin{rawhtml}
<!-- l2h_end sing_l2h 279 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 280 -->
\end{rawhtml}
$X$
\begin{rawhtml}
<!-- l2h_end sing_l2h 280 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 281 -->
\end{rawhtml}
$< X >$
\begin{rawhtml}
<!-- l2h_end sing_l2h 281 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 282 -->
\end{rawhtml}
$u < v$
\begin{rawhtml}
<!-- l2h_end sing_l2h 282 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 283 -->
\end{rawhtml}
$aub < avb$
\begin{rawhtml}
<!-- l2h_end sing_l2h 283 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 284 -->
\end{rawhtml}
$u$
\begin{rawhtml}
<!-- l2h_end sing_l2h 284 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 285 -->
\end{rawhtml}
$v$
\begin{rawhtml}
<!-- l2h_end sing_l2h 285 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 286 -->
\end{rawhtml}
$<X>$
\begin{rawhtml}
<!-- l2h_end sing_l2h 286 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 287 -->
\end{rawhtml}
$1 <= m$
\begin{rawhtml}
<!-- l2h_end sing_l2h 287 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 288 -->
\end{rawhtml}
$x_1> x_2 > $
\begin{rawhtml}
<!-- l2h_end sing_l2h 288 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 289 -->
\end{rawhtml}
$ > x_n$
\begin{rawhtml}
<!-- l2h_end sing_l2h 289 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 290 -->
\end{rawhtml}
$a<b$
\begin{rawhtml}
<!-- l2h_end sing_l2h 290 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 291 -->
\end{rawhtml}
$\exists u\in \langle X\rangle\setminus\{1\}:\;au=b$ or
\begin{rawhtml}
<!-- l2h_end sing_l2h 291 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 292 -->
\end{rawhtml}
$\exists u,\,v,\,w\in \langle X\rangle\;\exists x_i,\,x_j\in X: a=ux_iv,\;b=ux_jw$ and $i<j$ holds.
\begin{rawhtml}
<!-- l2h_end sing_l2h 292 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 293 -->
\end{rawhtml}
$(N_0,+)$
\begin{rawhtml}
<!-- l2h_end sing_l2h 293 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 294 -->
\end{rawhtml}
$w: \langle X\rangle\rightarrow N_0$
\begin{rawhtml}
<!-- l2h_end sing_l2h 294 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 295 -->
\end{rawhtml}
$w(x_i)=w_i$
\begin{rawhtml}
<!-- l2h_end sing_l2h 295 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 296 -->
\end{rawhtml}
$N_0$
\begin{rawhtml}
<!-- l2h_end sing_l2h 296 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 297 -->
\end{rawhtml}
$1 <= i <= n$
\begin{rawhtml}
<!-- l2h_end sing_l2h 297 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 298 -->
\end{rawhtml}
len:$ \langle X\rangle\rightarrow N_0$
\begin{rawhtml}
<!-- l2h_end sing_l2h 298 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 299 -->
\end{rawhtml}
$len(x_i)=1$
\begin{rawhtml}
<!-- l2h_end sing_l2h 299 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 300 -->
\end{rawhtml}
$w$
\begin{rawhtml}
<!-- l2h_end sing_l2h 300 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 301 -->
\end{rawhtml}
$<<$
\begin{rawhtml}
<!-- l2h_end sing_l2h 301 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 302 -->
\end{rawhtml}
$w(a)<w(b)$
\begin{rawhtml}
<!-- l2h_end sing_l2h 302 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 303 -->
\end{rawhtml}
$w(a)=w(b)$
\begin{rawhtml}
<!-- l2h_end sing_l2h 303 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 304 -->
\end{rawhtml}
$a<<b$
\begin{rawhtml}
<!-- l2h_end sing_l2h 304 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 305 -->
\end{rawhtml}
$\emptyset\not=Y\subset X$ if for all $f\in K\langle X\rangle\setminus\{0\}$ one has $ lm(f)\in  K\langle X\setminus Y\rangle\Rightarrow f\in K\langle X\setminus Y\rangle\subseteq K\langle X\rangle$.
\begin{rawhtml}
<!-- l2h_end sing_l2h 305 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 306 -->
\end{rawhtml}
$x_2$
\begin{rawhtml}
<!-- l2h_end sing_l2h 306 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 307 -->
\end{rawhtml}
$x_3>$
\begin{rawhtml}
<!-- l2h_end sing_l2h 307 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 308 -->
\end{rawhtml}
$x_1 x_1 x_1$
\begin{rawhtml}
<!-- l2h_end sing_l2h 308 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 309 -->
\end{rawhtml}
$x_3 x_2 x_1$
\begin{rawhtml}
<!-- l2h_end sing_l2h 309 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 310 -->
\end{rawhtml}
$x_1 x_2 x_3$
\begin{rawhtml}
<!-- l2h_end sing_l2h 310 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 311 -->
\end{rawhtml}
$x_3 x_3 x_3$
\begin{rawhtml}
<!-- l2h_end sing_l2h 311 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 312 -->
\end{rawhtml}
$x_3 x_1$
\begin{rawhtml}
<!-- l2h_end sing_l2h 312 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 313 -->
\end{rawhtml}
$x_2 x_2$
\begin{rawhtml}
<!-- l2h_end sing_l2h 313 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 314 -->
\end{rawhtml}
$x_1 x_3$
\begin{rawhtml}
<!-- l2h_end sing_l2h 314 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 315 -->
\end{rawhtml}
$x_2 x_3$
\begin{rawhtml}
<!-- l2h_end sing_l2h 315 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 316 -->
\end{rawhtml}
$x_3$
\begin{rawhtml}
<!-- l2h_end sing_l2h 316 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 317 -->
\end{rawhtml}
$x_1 x_1 x_1 > x_3 x_2 x_1 > x_1 x_2 x_3 > x_3 x_3 x_3 > x_3 x_1 > x_2 x_2 > x_1 x_3 > x_2 x_3 > x_1 > x_2 > x_3$
\begin{rawhtml}
<!-- l2h_end sing_l2h 317 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 318 -->
\end{rawhtml}
$x_1 x_1 x_1 > x_1 x_2 x_3 > x_3 x_2 x_1 > x_3 x_3 x_3 > x_1 x_3 > x_2 x_2 > x_2 x_3 > x_3 x_1 > x_1 > x_2 > x_3$
\begin{rawhtml}
<!-- l2h_end sing_l2h 318 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 319 -->
\end{rawhtml}
$w(x_i)=w_i>0$
\begin{rawhtml}
<!-- l2h_end sing_l2h 319 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 320 -->
\end{rawhtml}
$x_1 x_2 x_3 > x_2 x_2 > x_3 x_2 x_1 > x_1 x_1 x_1 > x_2 x_3 > x_3 x_3 x_3 > x_1 x_3 > x_2 > x_3 x_1 > x_1 > x_3$
\begin{rawhtml}
<!-- l2h_end sing_l2h 320 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 321 -->
\end{rawhtml}
Let $w^{(i)}$ be weights uniquely determined by $w^{(i)}(x_j)=\delta_{i,j}$ for $1\leq i,j\leq n$ where $\delta$ denotes the Kronecker delta. Let $<_n$ be the $w^{(n)}$-weight extension of the left lexicographical ordering on $\langle X\rangle$ and inductively $<_i$ be the $w^{(i)}$-weight extension of $<_{i+1}$ for all $1\leq i<n$.
The monomial ordering lp corresponds to $<_1$ and eliminates $x_1,...,x_j$ for all $1\leq j<n$.
We refer to it as to {\bf left elimination ordering}.
\begin{rawhtml}
<!-- l2h_end sing_l2h 321 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 322 -->
\end{rawhtml}
$<_1$
\begin{rawhtml}
<!-- l2h_end sing_l2h 322 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 323 -->
\end{rawhtml}
$x_j$
\begin{rawhtml}
<!-- l2h_end sing_l2h 323 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 324 -->
\end{rawhtml}
$x_1 x_1 x_1 > x_1 x_2 x_3 > x_3 x_2 x_1 > x_1 x_3 > x_3 x_1 > x_1 > x_2 x_2 > x_2 x_3 > x_2 > x_3 x_3 x_3 > x_3$
\begin{rawhtml}
<!-- l2h_end sing_l2h 324 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 325 -->
\end{rawhtml}
Let $w^{(i)}$ be weights uniquely determined by $w^{(i)}(x_j)=\delta_{i,j}$ for $1\leq i,j\leq n$ where $\delta$ denotes the Kronecker delta. Let $<_1$ be the $w^{(1)}$-weight extension of the left lexicographical ordering on $\langle X\rangle$ and inductively $<_i$ be the $w^{(i)}$-weight extension of $<_{i-1}$ for all $1<i\leq n$.
The monomial ordering \texttt{rp} corresponds to $<_n$ and eliminates $\{x_j,\ldots,x_n\}$ for all $1<j\leq n$.
We refer to it as to {\bf right elimination ordering}.
\begin{rawhtml}
<!-- l2h_end sing_l2h 325 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 326 -->
\end{rawhtml}
$<_n$
\begin{rawhtml}
<!-- l2h_end sing_l2h 326 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 327 -->
\end{rawhtml}
$1 < j <= n$
\begin{rawhtml}
<!-- l2h_end sing_l2h 327 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 328 -->
\end{rawhtml}
$x_3 x_3 x_3 > x_1 x_2 x_3 > x_3 x_2 x_1 > x_2 x_3 > x_1 x_3 > x_3 x_1 > x_3 > x_2 x_2 > x_2 > x_1 x_1 x_1 > x_1$
\begin{rawhtml}
<!-- l2h_end sing_l2h 328 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 329 -->
\end{rawhtml}
For weight $v:\langle X\rangle\rightarrow N_0$ determined by $v(x_i)=v_i\in N_0$ with $1\leq i\leq n$ and monomial ordering $\prec$ on $\langle X\rangle$, the $v$\emph{-weight extension} of $\prec$ corresponds to \texttt{(a(v), o)}. As a choice for $\prec$ there are currently two options implemented, which are \texttt{dp} and \texttt{Dp}. Notice that this ordering eliminates $\{x_i\in X\mid v(x_i)\not=0\}$.
\begin{rawhtml}
<!-- l2h_end sing_l2h 329 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 330 -->
\end{rawhtml}
$x_1 x_1 x_1 > x_1 x_2 x_3 > x_3 x_2 x_1 > x_1 x_3 > x_3 x_1 > x_1 > x_3 x_3 x_3 > x_2 x_2 > x_2 x_3 > x_2 > x_3$
\begin{rawhtml}
<!-- l2h_end sing_l2h 330 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 331 -->
\end{rawhtml}
$x_1 x_1 x_1 > x_1 x_2 x_3 > x_3 x_2 x_1 > x_2 x_2 > x_1 x_3 > x_2 x_3 > x_3 x_1 > x_1 > x_2 > x_3 x_3 x_3 > x_3$
\begin{rawhtml}
<!-- l2h_end sing_l2h 331 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 332 -->
\end{rawhtml}
$p,s \in X$, such that $w = p \cdot v \cdot s$,
\begin{rawhtml}
<!-- l2h_end sing_l2h 332 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 333 -->
\end{rawhtml}
$T := K\langle x_1,\dots,x_n \rangle$
\begin{rawhtml}
<!-- l2h_end sing_l2h 333 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 334 -->
\end{rawhtml}
For a subset $G \subset K\langle x_1,\dots,x_n \rangle$,
define the \textbf{leading ideal} of $G$ to be the two-sided ideal
$LM(G) \; = \; {}_{T} \langle$ $\; \{lm(g) \;|\; g \in G\setminus\{0\} \}$ $\; \rangle_{T} \subseteq T$.


A subset $G \subset I$ is a \textbf{(two-sided) Groebner basis} for the ideal $I$ with respect to $<$, if $LM(G) = LM(I)$.

That is $\forall f\in I\setminus\{0\}$ there exists $g\in G$, such that
$lm(g)$ divides $lm(f)$.
\begin{rawhtml}
<!-- l2h_end sing_l2h 334 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 335 -->
\end{rawhtml}
$G(d)$
\begin{rawhtml}
<!-- l2h_end sing_l2h 335 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 336 -->
\end{rawhtml}
$G(2d-1)=G(d)$
\begin{rawhtml}
<!-- l2h_end sing_l2h 336 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 337 -->
\end{rawhtml}
$A = K$
\begin{rawhtml}
<!-- l2h_end sing_l2h 337 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 338 -->
\end{rawhtml}
$A e_1 A \oplus \ldots \oplus A e_r A$,
\begin{rawhtml}
<!-- l2h_end sing_l2h 338 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 339 -->
\end{rawhtml}
$e_i$
\begin{rawhtml}
<!-- l2h_end sing_l2h 339 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 340 -->
\end{rawhtml}
$AeA$
\begin{rawhtml}
<!-- l2h_end sing_l2h 340 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 341 -->
\end{rawhtml}
$N = \langle g_1,\ldots,g_m \rangle \subset \bigoplus_{i=1}^r A e_i A$
\begin{rawhtml}
<!-- l2h_end sing_l2h 341 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 342 -->
\end{rawhtml}
$
\bigoplus_{j=1}^m A \epsilon_j A \longrightarrow \bigoplus_{i=1}^r A e_i A,
\;
\epsilon_j  \mapsto g_j,
$
\begin{rawhtml}
<!-- l2h_end sing_l2h 342 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 343 -->
\end{rawhtml}
$\sum_{j=1}^m \sum_k  \ell_{jk} \epsilon_j r_{jk}
\mapsto \sum_{j=1}^m \sum_k \ell_{jk} g_j r_{jk}.
$
\begin{rawhtml}
<!-- l2h_end sing_l2h 343 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 344 -->
\end{rawhtml}
$N$
\begin{rawhtml}
<!-- l2h_end sing_l2h 344 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 345 -->
\end{rawhtml}
$N_1, \ldots, N_s$
\begin{rawhtml}
<!-- l2h_end sing_l2h 345 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 346 -->
\end{rawhtml}
$M_1, \ldots, M_m$
\begin{rawhtml}
<!-- l2h_end sing_l2h 346 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 347 -->
\end{rawhtml}
$N_i = \sum_{j=1}^m \sum_k \ell_{jk} M_j r_{jk} = \sum_{j=1}^m T_{ij} M_j,$
\begin{rawhtml}
<!-- l2h_end sing_l2h 347 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 348 -->
\end{rawhtml}
$T_ij$
\begin{rawhtml}
<!-- l2h_end sing_l2h 348 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 349 -->
\end{rawhtml}
$R \langle X \rangle \otimes R \langle X \rangle,$
\begin{rawhtml}
<!-- l2h_end sing_l2h 349 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 350 -->
\end{rawhtml}
Already Feynman and Rota encoded
the monomials (words) of the free algebra
$x_{i_1} x_{i_2} \dots x_{i_m} \in K\langle x_1,\ldots,x_n \rangle$
via the double-indexed letterplace
(that is encoding the letter (= variable) and its place in the word) monomials
$x(i_1 | 1) x(i_2 | 2) \dots x(i_m | m) \in K[X\times N]$, where $X=\{x_1,\ldots,x_n\}$
and $N$ is the semigroup of natural numbers, starting with 1 as the first possible place.
Note, that the letterplace algebra $K[X \times N]$ is an infinitely generated commutative polynomial $K$-algebra.
Since $K\langle x_1,\ldots,x_n \rangle$ is not Noetherian, it is common to perform the computations with
its ideals and modules up to a given degree bound.
\begin{rawhtml}
<!-- l2h_end sing_l2h 350 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 351 -->
\end{rawhtml}
Subject to the given degree (length) bound $d$, the truncated letterplace algebra
$K[X\times (1, ..., d)]$ is finitely generated commutative polynomial $K$-algebra.
\begin{rawhtml}
<!-- l2h_end sing_l2h 351 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 352 -->
\end{rawhtml}
$K$-algebra operations of the letterplace algebra \textbf{and} simultaneous shifting of places by any natural number $n$.
\begin{rawhtml}
<!-- l2h_end sing_l2h 352 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 353 -->
\end{rawhtml}
$R<$
\begin{rawhtml}
<!-- l2h_end sing_l2h 353 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 354 -->
\end{rawhtml}
$>$
\begin{rawhtml}
<!-- l2h_end sing_l2h 354 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 355 -->
\end{rawhtml}
$R=Z$
\begin{rawhtml}
<!-- l2h_end sing_l2h 355 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 356 -->
\end{rawhtml}
That is, choosing $>$ implements the ring
$$
  K[x]_> := \left\{{{f}\over{u}}\; \bigg|\; f, u \in K[x],
  \, LM(u) =
  1\right\}.
$$
\begin{rawhtml}
<!-- l2h_end sing_l2h 356 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 357 -->
\end{rawhtml}
$dim_{Q(t)}Q(t)[x,y]/j$.
\begin{rawhtml}
<!-- l2h_end sing_l2h 357 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 358 -->
\end{rawhtml}
$a \in Q$
\begin{rawhtml}
<!-- l2h_end sing_l2h 358 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 359 -->
\end{rawhtml}
$dim_Q Q[x,y]/j_0$,
\begin{rawhtml}
<!-- l2h_end sing_l2h 359 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 360 -->
\end{rawhtml}
$j_0=j|_{t=a}$.
\begin{rawhtml}
<!-- l2h_end sing_l2h 360 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 361 -->
\end{rawhtml}
$I\subset K[x_1,\dots,x_n]$
\begin{rawhtml}
<!-- l2h_end sing_l2h 361 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 362 -->
\end{rawhtml}
$x_0$
\begin{rawhtml}
<!-- l2h_end sing_l2h 362 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 363 -->
\end{rawhtml}
$K[x_1,\dots,x_n]$
\begin{rawhtml}
<!-- l2h_end sing_l2h 363 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 364 -->
\end{rawhtml}
$K[x_0,\dots,x_n]$
\begin{rawhtml}
<!-- l2h_end sing_l2h 364 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 365 -->
\end{rawhtml}
$i, j$
\begin{rawhtml}
<!-- l2h_end sing_l2h 365 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 366 -->
\end{rawhtml}
$$
\hbox{sat}(i,j)=\{x\in R\;|\; \exists\;n\hbox{ s.t. }
x\cdot(j^n)\subseteq i\}
= \bigcup_{n=1}^\infty i:j^n$$
\begin{rawhtml}
<!-- l2h_end sing_l2h 366 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 367 -->
\end{rawhtml}
$V(i)$
\begin{rawhtml}
<!-- l2h_end sing_l2h 367 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 368 -->
\end{rawhtml}
$sat(j+(f),m)$
\begin{rawhtml}
<!-- l2h_end sing_l2h 368 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 369 -->
\end{rawhtml}
$p^k$
\begin{rawhtml}
<!-- l2h_end sing_l2h 369 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 370 -->
\end{rawhtml}
$f=(f_1,\ldots,f_n):k^r\rightarrow k^n$
\begin{rawhtml}
<!-- l2h_end sing_l2h 370 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 371 -->
\end{rawhtml}
$j=J \cap k[x_1,\ldots,x_n]$,
\begin{rawhtml}
<!-- l2h_end sing_l2h 371 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 372 -->
\end{rawhtml}
$$J=(x_1-f_1(t_1,\ldots,t_r),\ldots,x_n-f_n(t_1,\ldots,t_r))\subseteq
k[t_1,\ldots,t_r,x_1,\ldots,x_n]$$
\begin{rawhtml}
<!-- l2h_end sing_l2h 372 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 373 -->
\end{rawhtml}
$t_1,\ldots,t_r$.
\begin{rawhtml}
<!-- l2h_end sing_l2h 373 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 374 -->
\end{rawhtml}
$f:(k^r,0)\rightarrow(k^n,0)$,
\begin{rawhtml}
<!-- l2h_end sing_l2h 374 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 375 -->
\end{rawhtml}
$k^r\times(k^n,0)$,
\begin{rawhtml}
<!-- l2h_end sing_l2h 375 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 376 -->
\end{rawhtml}
$$\hbox{pr}:k^r\times(k^n,0)\rightarrow(k^n,0)$$
can be computed.
\begin{rawhtml}
<!-- l2h_end sing_l2h 376 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 377 -->
\end{rawhtml}
$P^4$.
\begin{rawhtml}
<!-- l2h_end sing_l2h 377 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 378 -->
\end{rawhtml}
$K^3$
\begin{rawhtml}
<!-- l2h_end sing_l2h 378 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 379 -->
\end{rawhtml}
$K^3$.
\begin{rawhtml}
<!-- l2h_end sing_l2h 379 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 380 -->
\end{rawhtml}
$P^4$
\begin{rawhtml}
<!-- l2h_end sing_l2h 380 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 381 -->
\end{rawhtml}
$$F=R(-2)^2\oplus R(-1)^2\oplus R\,.$$
\begin{rawhtml}
<!-- l2h_end sing_l2h 381 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 382 -->
\end{rawhtml}
$$A/(A \cap B)$$
\begin{rawhtml}
<!-- l2h_end sing_l2h 382 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 383 -->
\end{rawhtml}
$\hbox{Ext}^1(M,M)$, resp.\ $\hbox{Ext}^2(M,M)$,
\begin{rawhtml}
<!-- l2h_end sing_l2h 383 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 384 -->
\end{rawhtml}
$\hbox{Ext}^1$
\begin{rawhtml}
<!-- l2h_end sing_l2h 384 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 385 -->
\end{rawhtml}
$\hbox{Ext}^1$, $\hbox{Ext}^2$
\begin{rawhtml}
<!-- l2h_end sing_l2h 385 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 386 -->
\end{rawhtml}
$\hbox{Ext}^k(R/J,R)$
\begin{rawhtml}
<!-- l2h_end sing_l2h 386 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 387 -->
\end{rawhtml}
 ($=\hbox{Ext}^1(K,K)$)
\begin{rawhtml}
<!-- l2h_end sing_l2h 387 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 388 -->
\end{rawhtml}
 ($=\hbox{Ext}^2(K,K)$)
\begin{rawhtml}
<!-- l2h_end sing_l2h 388 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 389 -->
\end{rawhtml}
$K=R/m$
\begin{rawhtml}
<!-- l2h_end sing_l2h 389 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 390 -->
\end{rawhtml}
$R=K[x,y]_m/(x^2-y^3)$, $m=(x,y)$.
\begin{rawhtml}
<!-- l2h_end sing_l2h 390 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 391 -->
\end{rawhtml}
$\hbox{Ext}^1(m,m)$
\begin{rawhtml}
<!-- l2h_end sing_l2h 391 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 392 -->
\end{rawhtml}
$\hbox{Ext}^2(K,K)$.
\begin{rawhtml}
<!-- l2h_end sing_l2h 392 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 393 -->
\end{rawhtml}
$\hbox{Ext}^k(R/i,R)$
\begin{rawhtml}
<!-- l2h_end sing_l2h 393 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 394 -->
\end{rawhtml}
D$_k$(R)
\begin{rawhtml}
<!-- l2h_end sing_l2h 394 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 395 -->
\end{rawhtml}
$(m+1)$
\begin{rawhtml}
<!-- l2h_end sing_l2h 395 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 396 -->
\end{rawhtml}
$(n \times n)$-matrix.
\begin{rawhtml}
<!-- l2h_end sing_l2h 396 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 397 -->
\end{rawhtml}
$\hbox{depth}(\hbox{D}_k(R))\geq m(m+1)/2 + m-1$
\begin{rawhtml}
<!-- l2h_end sing_l2h 397 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 398 -->
\end{rawhtml}
$R/I$
\begin{rawhtml}
<!-- l2h_end sing_l2h 398 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 399 -->
\end{rawhtml}
$m\times r$ and $m\times s$
\begin{rawhtml}
<!-- l2h_end sing_l2h 399 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 400 -->
\end{rawhtml}
$$
R^r \buildrel{A}\over{\longrightarrow}
R^m \buildrel{B}\over{\longleftarrow} R^s\;.
$$
\begin{rawhtml}
<!-- l2h_end sing_l2h 400 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 401 -->
\end{rawhtml}
$R^r \buildrel{A}\over{\longrightarrow}
R^m\longrightarrow
R^m/\hbox{Im}(B) \;.$
\begin{rawhtml}
<!-- l2h_end sing_l2h 401 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 402 -->
\end{rawhtml}
$$
\hbox{\tt modulo}(A,B)=\hbox{ker}(R^r
\buildrel{A}\over{\longrightarrow}R^m/\hbox{Im}(B)) \; .
$$
\begin{rawhtml}
<!-- l2h_end sing_l2h 402 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 403 -->
\end{rawhtml}
$g$, $f_1$, \dots, $f_r\in K[x_1,\ldots,x_n]$.
\begin{rawhtml}
<!-- l2h_end sing_l2h 403 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 404 -->
\end{rawhtml}
$f_1$, \dots, $f_r$
\begin{rawhtml}
<!-- l2h_end sing_l2h 404 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 405 -->
\end{rawhtml}
$I=\langle Y_1-f_1,\ldots,Y_r-f_r \rangle \subseteq
K[x_1,\ldots,x_n,Y_1,\ldots,Y_r]$.
\begin{rawhtml}
<!-- l2h_end sing_l2h 405 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 406 -->
\end{rawhtml}
$I \cap K[Y_1,\ldots,Y_r]$
\begin{rawhtml}
<!-- l2h_end sing_l2h 406 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 407 -->
\end{rawhtml}
$f_1$, \dots, $f_r$.
\begin{rawhtml}
<!-- l2h_end sing_l2h 407 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 408 -->
\end{rawhtml}
$g \in K [f_1,\ldots,f_r]$.
\begin{rawhtml}
<!-- l2h_end sing_l2h 408 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 409 -->
\end{rawhtml}
$g \in K[f_1,\ldots,f_r]$
\begin{rawhtml}
<!-- l2h_end sing_l2h 409 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 410 -->
\end{rawhtml}
$g$
\begin{rawhtml}
<!-- l2h_end sing_l2h 410 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 411 -->
\end{rawhtml}
$X=(x_1,\ldots,x_n)$ and $Y=(Y_1,\ldots,Y_r)$ with $X>Y$
\begin{rawhtml}
<!-- l2h_end sing_l2h 411 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 412 -->
\end{rawhtml}
$K[Y]$
\begin{rawhtml}
<!-- l2h_end sing_l2h 412 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 413 -->
\end{rawhtml}
$K[[x_1,\ldots,x_n]]$
\begin{rawhtml}
<!-- l2h_end sing_l2h 413 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 414 -->
\end{rawhtml}
$$
\hbox{milnor}(f) = \hbox{dim}_K(K[[x_1,\ldots,x_n]]/\hbox{jacob}(f)),
$$
respectively
$$
\hbox{tjurina}(f) = \hbox{dim}_K(K[[x_1,\ldots,x_n]]/((f)+\hbox{jacob}(f)))
$$
where
\begin{rawhtml}
<!-- l2h_end sing_l2h 414 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 415 -->
\end{rawhtml}
$\hbox{Loc}_{(x)}K[x_1,\ldots,x_n]$,
\begin{rawhtml}
<!-- l2h_end sing_l2h 415 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 416 -->
\end{rawhtml}
$K[x_1,\ldots,x_n]$
\begin{rawhtml}
<!-- l2h_end sing_l2h 416 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 417 -->
\end{rawhtml}
$(x_1,\ldots,x_n)$.
\begin{rawhtml}
<!-- l2h_end sing_l2h 417 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 418 -->
\end{rawhtml}
$\hbox{Loc}_{(x)}K[x_1,\ldots,x_n])$
\begin{rawhtml}
<!-- l2h_end sing_l2h 418 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 419 -->
\end{rawhtml}
$\hbox{dim}_K(\hbox{Loc}_{(x,y,z)}K[x,y,z]/\hbox{jacob}(f))$
\begin{rawhtml}
<!-- l2h_end sing_l2h 419 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 420 -->
\end{rawhtml}
$\hbox{dim}_K(K[x,y,z]/\hbox{jacob}(f))$
\begin{rawhtml}
<!-- l2h_end sing_l2h 420 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 421 -->
\end{rawhtml}
$\hbox{dim}_K(\hbox{Loc}_{(x,y,z)}K[x,y,z]/(\hbox{jacob}(f)+(f)))$
\begin{rawhtml}
<!-- l2h_end sing_l2h 421 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 422 -->
\end{rawhtml}
$\hbox{dim}_K(K[x,y,z]/(\hbox{jacob}(f)+(f)))$,
\begin{rawhtml}
<!-- l2h_end sing_l2h 422 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 423 -->
\end{rawhtml}
$f\in k[x_1,\ldots,x_n,t]$
\begin{rawhtml}
<!-- l2h_end sing_l2h 423 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 424 -->
\end{rawhtml}
$t$
\begin{rawhtml}
<!-- l2h_end sing_l2h 424 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 425 -->
\end{rawhtml}
$f=0$
\begin{rawhtml}
<!-- l2h_end sing_l2h 425 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 426 -->
\end{rawhtml}
$V(\partial f/\partial x_1,\ldots,\partial f/\partial x_n) \setminus V(f)$
\begin{rawhtml}
<!-- l2h_end sing_l2h 426 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 427 -->
\end{rawhtml}
$V(\partial f/\partial x_1,\ldots,\partial f/\partial x_n)$
\begin{rawhtml}
<!-- l2h_end sing_l2h 427 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 428 -->
\end{rawhtml}
(in $P^n$),
\begin{rawhtml}
<!-- l2h_end sing_l2h 428 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 429 -->
\end{rawhtml}
(in $k^n$)
\begin{rawhtml}
<!-- l2h_end sing_l2h 429 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 430 -->
\end{rawhtml}
(in $(k^n,0)$),
\begin{rawhtml}
<!-- l2h_end sing_l2h 430 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 431 -->
\end{rawhtml}
$T^1$
\begin{rawhtml}
<!-- l2h_end sing_l2h 431 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 432 -->
\end{rawhtml}
$T^2$
\begin{rawhtml}
<!-- l2h_end sing_l2h 432 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 433 -->
\end{rawhtml}
$T^1,T_2$
\begin{rawhtml}
<!-- l2h_end sing_l2h 433 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 434 -->
\end{rawhtml}
$T^2=0$
\begin{rawhtml}
<!-- l2h_end sing_l2h 434 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 435 -->
\end{rawhtml}
$h_1,\ldots,h_r$
\begin{rawhtml}
<!-- l2h_end sing_l2h 435 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 436 -->
\end{rawhtml}
$I \subset R$ is the ideal generated by $f_1,...,f_s$, then any infinitesimal
deformation of $R/I$ over $K[\varepsilon]/(\varepsilon^2)$ is given
by $f+\varepsilon g$,
where $f=(f_1,...,f_s)$, and where $g$ is a $K$-linear combination of
the $h_i$.
\begin{rawhtml}
<!-- l2h_end sing_l2h 436 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 437 -->
\end{rawhtml}
$\sum x_i \partial /\partial x_{i+1}$.
\begin{rawhtml}
<!-- l2h_end sing_l2h 437 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 438 -->
\end{rawhtml}
$A:=U(sl_2)$
\begin{rawhtml}
<!-- l2h_end sing_l2h 438 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 439 -->
\end{rawhtml}
$I = \{ e^2, f \}$.
\begin{rawhtml}
<!-- l2h_end sing_l2h 439 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 440 -->
\end{rawhtml}
$[f_1,...,f_n]$ or $f_1*gen(1)+...+f_n*gen(n)$, where $gen(i)$
\begin{rawhtml}
<!-- l2h_end sing_l2h 440 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 441 -->
\end{rawhtml}
$gen(i)$
\begin{rawhtml}
<!-- l2h_end sing_l2h 441 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 442 -->
\end{rawhtml}
$[f_1,...,f_n]$
\begin{rawhtml}
<!-- l2h_end sing_l2h 442 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 443 -->
\end{rawhtml}
$v=[f_1,...,f_n]$
\begin{rawhtml}
<!-- l2h_end sing_l2h 443 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 444 -->
\end{rawhtml}
nrows($v$)
\begin{rawhtml}
<!-- l2h_end sing_l2h 444 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 445 -->
\end{rawhtml}
$f_r \not= 0$.
\begin{rawhtml}
<!-- l2h_end sing_l2h 445 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 446 -->
\end{rawhtml}
$f_i*gen(i)$
\begin{rawhtml}
<!-- l2h_end sing_l2h 446 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 447 -->
\end{rawhtml}
$f_i=0$
\begin{rawhtml}
<!-- l2h_end sing_l2h 447 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 448 -->
\end{rawhtml}
$v_1,...,v_k$
\begin{rawhtml}
<!-- l2h_end sing_l2h 448 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 449 -->
\end{rawhtml}
nrows($M$)
\begin{rawhtml}
<!-- l2h_end sing_l2h 449 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 450 -->
\end{rawhtml}
nrows($v_i$).
\begin{rawhtml}
<!-- l2h_end sing_l2h 450 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 451 -->
\end{rawhtml}
$R$.
\begin{rawhtml}
<!-- l2h_end sing_l2h 451 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 452 -->
\end{rawhtml}
$N$,
\begin{rawhtml}
<!-- l2h_end sing_l2h 452 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 453 -->
\end{rawhtml}
$N = R^n/M$ where $n$
\begin{rawhtml}
<!-- l2h_end sing_l2h 453 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 454 -->
\end{rawhtml}
$N$ and $M \subseteq R^n$
\begin{rawhtml}
<!-- l2h_end sing_l2h 454 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 455 -->
\end{rawhtml}
$N = R^n/M$.
\begin{rawhtml}
<!-- l2h_end sing_l2h 455 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 456 -->
\end{rawhtml}
$M$,
\begin{rawhtml}
<!-- l2h_end sing_l2h 456 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 457 -->
\end{rawhtml}
$N = R^n/M$ instead of $M$.
\begin{rawhtml}
<!-- l2h_end sing_l2h 457 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 458 -->
\end{rawhtml}
$N$ as $N = R^n/$std($M$)).
\begin{rawhtml}
<!-- l2h_end sing_l2h 458 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 459 -->
\end{rawhtml}
dim$(R^n/M)$, resp.@: dim$_k(R^n/M)$
\begin{rawhtml}
<!-- l2h_end sing_l2h 459 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 460 -->
\end{rawhtml}
$N$.
\begin{rawhtml}
<!-- l2h_end sing_l2h 460 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 461 -->
\end{rawhtml}
$N = R^n/M$
\begin{rawhtml}
<!-- l2h_end sing_l2h 461 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 462 -->
\end{rawhtml}
$\hbox{Loc}_{(x)}K[x_1,\ldots,x_n]$.
\begin{rawhtml}
<!-- l2h_end sing_l2h 462 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 463 -->
\end{rawhtml}
$\hbox{Wp}(w_1, \ldots, w_n)$
\begin{rawhtml}
<!-- l2h_end sing_l2h 463 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 464 -->
\end{rawhtml}
$\hbox{Ws}(w_1, \ldots, w_n)$)
\begin{rawhtml}
<!-- l2h_end sing_l2h 464 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 465 -->
\end{rawhtml}
$w_1$, $\ldots$, $w_n$ of $x_1$, $\ldots$, $x_n$.
\begin{rawhtml}
<!-- l2h_end sing_l2h 465 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 466 -->
\end{rawhtml}
A monomial ordering (term ordering) on $K[x_1, \ldots, x_n]$ is
a total ordering $<$ on the
set of monomials (power products) $\{x^\alpha \mid \alpha \in \bf{N}^n\}$
which is compatible with the
natural semigroup structure, i.e., $x^\alpha < x^\beta$ implies $x^\gamma
x^\alpha < x^\gamma x^\beta$ for any $\gamma \in \bf{N}^n$.
We do not require
$<$ to be  a wellordering.
\begin{rawhtml}
<!-- l2h_end sing_l2h 466 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 467 -->
\end{rawhtml}
$M$ in $GL(n,R)$,
\begin{rawhtml}
<!-- l2h_end sing_l2h 467 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 468 -->
\end{rawhtml}
Global orderings are wellorderings (i.e.,  \hbox{$1 < x_i$} for each variable
$x_i$), local orderings satisfy $1 > x_i$ for each variable.   If some variables are ordered globally and others locally we
call it a mixed ordering.   Local or mixed orderings are not wellorderings.

Let $K$ be the ground field, \hbox{$x = (x_1, \ldots, x_n)$} the
variables and $<$ a monomial ordering, then Loc $K[x]$ denotes the
localization of $K[x]$ with respect to the multiplicatively closed set $$\{1 +
g \mid g = 0 \hbox{ or } g \in K[x]\backslash \{0\} \hbox{ and }L(g) <
1\}.$$   Here, $L(g)$
denotes the leading monomial of $g$, i.e., the biggest monomial of $g$ with
respect to $<$.   The result of any computation which uses standard basis
computations has to be interpreted in Loc $K[x]$.
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For all these orderings, we have Loc $K[x]$ = $K[x]$
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$x^\alpha < x^\beta  \Leftrightarrow  \exists\; 1 \le i \le n :
\alpha_1 = \beta_1, \ldots, \alpha_{i-1} = \beta_{i-1}, \alpha_i <
\beta_i$.
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$x^\alpha < x^\beta  \Leftrightarrow  \exists\; 1 \le i \le n :
\alpha_n = \beta_n,
    \ldots, \alpha_{i+1} = \beta_{i+1}, \alpha_i < \beta_i.$
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let $\deg(x^\alpha) = \alpha_1 + \cdots + \alpha_n,$ then
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    $x^\alpha < x^\beta \Leftrightarrow \deg(x^\alpha) < \deg(x^\beta)$ or
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    \phantom{$x^\alpha < x^\beta \Leftrightarrow $}$ \deg(x^\alpha) =
    \deg(x^\beta)$ and $\exists\ 1 \le i \le n: \alpha_n = \beta_n,
    \ldots, \alpha_{i+1} = \beta_{i+1}, \alpha_i > \beta_i.$
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    \phantom{ $x^\alpha < x^\beta \Leftrightarrow $} $\deg(x^\alpha) =
    \deg(x^\beta)$ and $\exists\ 1 \le i \le n:\alpha_1 = \beta_1,
    \ldots, \alpha_{i-1} = \beta_{i-1}, \alpha_i < \beta_i.$
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let $w_1, \ldots, w_n$ be positive integers. Then ${\tt wp}(w_1, \ldots,
w_n)$
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$\deg(x^\alpha) = w_1 \alpha_1 + \cdots + w_n\alpha_n.$
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let $w_1, \ldots, w_n$ be positive integers. Then ${\tt Wp}(w_1, \ldots,
w_n)$
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Loc $K[x]$ = $K[x]_{(x)}$,
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$K[x]$
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\ $(x) = (x_1, ..., x_n)$.
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$x^\alpha < x^\beta  \Leftrightarrow  \exists\; 1 \le i \le n :
\alpha_1 = \beta_1, \ldots, \alpha_{i-1} = \beta_{i-1}, \alpha_i >
\beta_i$.
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    $x^\alpha < x^\beta \Leftrightarrow \deg(x^\alpha) > \deg(x^\beta)$ or
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    \phantom{ $x^\alpha < x^\beta \Leftrightarrow$}$ \deg(x^\alpha) =
    \deg(x^\beta)$ and $\exists\ 1 \le i \le n: \alpha_n = \beta_n,
    \ldots, \alpha_{i+1} = \beta_{i+1}, \alpha_i > \beta_i.$
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    \phantom{ $ x^\alpha < x^\beta \Leftrightarrow$}$ \deg(x^\alpha) =
    \deg(x^\beta)$ and $\exists\ 1 \le i \le n:\alpha_1 = \beta_1,
    \ldots, \alpha_{i-1} = \beta_{i-1}, \alpha_i < \beta_i.$
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${\tt ws}(w_1, \ldots, w_n),\; w_1$
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$w_2,\ldots,w_n$
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${\tt Ws}(w_1, \ldots, w_n),\; w_1$
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$\{ x^a e_i  \mid  a \in N^n, 1 \leq i \leq r \}$ in Loc $K[x]^r$ = Loc
$K[x]e_1
+ \ldots +$Loc $K[x]e_r$, where $e_1, \ldots, e_r$ denote the canonical
generators of Loc $K[x]^r$, the r-fold direct sum of Loc $K[x]$.
(The function {\tt gen(i)} yields $e_i$).
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 Loc $K[x]^r$
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Loc $K[x]$
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$<_m = (<,C)$ denotes the module ordering (giving priority to the coefficients):
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\quad  \quad  $x^\alpha e_i <_m x^\beta e_j \Leftrightarrow x^\alpha <
x^\beta$ or ($x^\alpha = x^\beta $ and $ i < j$).
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$<_m = (C, <)$ denotes the module ordering (giving priority to the component):
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\quad \quad   $x^\alpha e_i <_m x^\beta e_j \Leftrightarrow i < j$ or ($
i = j $ and $ x^\alpha < x^\beta $).
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$<_m = (<,c)$ denotes the module ordering (giving priority to the coefficients):
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\quad \quad $x^\alpha e_i <_m x^\beta e_j \Leftrightarrow x^\alpha <
x^\beta$ or ($x^\alpha = x^\beta $ and $ i > j$).
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$<_m = (c, <)$ denotes the module ordering (giving priority to the component):
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\quad \quad   $x^\alpha e_i <_m x^\beta e_j \Leftrightarrow i > j$ or ($
i = j $ and $ x^\alpha < x^\beta $).
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The output of a vector $v$ in $K[x]^r$ with components $v_1,
\ldots, v_r$ has the format $v_1 * gen(1) + \ldots + v_r * gen(r)$
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In this case a vector is written as $[v_1, \ldots, v_r]$.
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$(n \times n)$-matrix
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$M_1, \ldots, M_n$ the rows of $M$.
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\quad \quad $x^a < x^b \Leftrightarrow \exists\  1 \leq i \leq n :
M_1 a = \; M_1 b, \ldots, M_{i-1} a = \; M_{i-1} b$ and $M_i a < \; M_i b$.
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$x^a < x^b$
if and only if $M a$ is smaller than $M b$
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$\quad$ lp:
$\left(\matrix{
 1 & 0 & 0 \cr
 0 & 1 & 0 \cr
 0 & 0 & 1 \cr
 }\right)$
\quad dp:
$\left(\matrix{
 1 & 1 & 1 \cr
 0 & 0 &-1 \cr
 0 &-1 & 0 \cr
 }\right)$
\quad Dp:
$\left(\matrix{
 1 & 1 & 1 \cr
 1 & 0 & 0 \cr
 0 & 1 & 0 \cr
 }\right)$

$\quad$ wp(1,2,3):
$\left(\matrix{
 1 & 2 & 3 \cr
 0 & 0 &-1 \cr
 0 &-1 & 0 \cr
 }\right)$
\quad Wp(1,2,3):
$\left(\matrix{
 1 & 2 & 3 \cr
 1 & 0 & 0 \cr
 0 & 1 & 0 \cr
 }\right)$

$\quad$ ls:
$\left(\matrix{
-1 & 0 & 0 \cr
 0 &-1 & 0 \cr
 0 & 0 &-1 \cr
 }\right)$
\quad ds:
$\left(\matrix{
-1 &-1 &-1 \cr
 0 & 0 &-1 \cr
 0 &-1 & 0 \cr
 }\right)$
\quad Ds:
$\left(\matrix{
-1 &-1 &-1 \cr
 1 & 0 & 0 \cr
 0 & 1 & 0 \cr
 }\right)$

$\quad$ ws(1,2,3):
$\left(\matrix{
-1 &-2 &-3 \cr
 0 & 0 &-1 \cr
 0 &-1 & 0 \cr
 }\right)$
\quad Ws(1,2,3):
$\left(\matrix{
-1 &-2 &-3 \cr
 1 & 0 & 0 \cr
 0 & 1 & 0 \cr
 }\right)$
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$\quad$ (dp(3), wp(1,2,3)):
$\left(\matrix{
1&  1&  1&  0&  0&  0 \cr
0&  0&  -1&  0&  0&  0 \cr
0&  -1&  0&  0&  0&  0 \cr
0&  0&  0&  1&  2&  3 \cr
0&  0&  0&  0&  0&  -1 \cr
0&  0&  0&  0&  -1&  0 \cr
 }\right)$

$\quad$ (Dp(3), ds(3)):
$\left(\matrix{
1&  1&  1&  0&  0&  0 \cr
1&  0&  0&  0&  0&  0 \cr
0&  1&  0&  0&  0&  0 \cr
0&  0&  0&  -1&  -1&  -1 \cr
0&  0&  0&  0&  0&  -1 \cr
0&  0&  0&  0&  -1&  0 \cr
 }\right)$
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$\quad$ (dp(3), a(1,2,3),dp(3)):
$\left(\matrix{
1&  1&  1&  0&  0&  0 \cr
0&  0&  -1&  0&  0&  0 \cr
0&  -1&  0&  0&  0&  0 \cr
0&  0&  0&  1&  2&  3 \cr
0&  0&  0&  1&  1&  1 \cr
0&  0&  0&  0&  0&  -1 \cr
0&  0&  0&  0&  -1&  0 \cr
 }\right)$

$\quad$ (a(1,2,3,4,5),Dp(3), ds(3)):
$\left(\matrix{
1&  2&  3&  4&  5&  0 \cr
1&  1&  1&  0&  0&  0 \cr
1&  0&  0&  0&  0&  0 \cr
0&  1&  0&  0&  0&  0 \cr
0&  0&  0&  -1&  -1&  -1 \cr
0&  0&  0&  0&  0 & -1 \cr
0&  0&  0&  0&  -1&  0 \cr
 }\right)$
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$n \times n$
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$x = (x_1, \ldots, x_n)$ and $y = (y_1, \ldots, y_m)$
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$<_1$ a monomial
ordering on $K[x]$ and $<_2$ a monomial ordering on $K[y]$.   The product
ordering (or block ordering) $<\ := (<_1,<_2)$ on $K[x,y]$ is the following:
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\quad \quad $x^a y^b < x^A y^B \Leftrightarrow x^a <_1 x^A $ or ($x^a =
x^A$ and $y^b <_2 y^B$).
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${\tt a}(w_1, \ldots, w_n),\; $
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$w_1,\ldots,w_n$
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$\deg(x^\alpha) = w_1 \alpha_1 + \cdots + w_n\alpha_n$
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    $$\deg(x^\alpha) < \deg(x^\beta) \Rightarrow x^\alpha < x^\beta,$$
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    $$\deg(x^\alpha) > \deg(x^\beta) \Rightarrow x^\alpha > x^\beta. $$
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Let $R = \hbox{Loc}_< K[\underline{x}]$ and let $I$ be a submodule of $R^r$.
Note that for r=1 this means that $I$ is an ideal in $R$.
Denote by $L(I)$ the submodule of $R^r$ generated by the leading terms
of elements of $I$, i.e. by $\left\{L(f) \mid f \in I\right\}$.
Then $f_1, \ldots, f_s \in I$ is called a {\bf standard basis} of $I$
if $L(f_1), \ldots, L(f_s)$ generate $L(I)$.

A standard basis is {\bf minimal} if $\forall i: (f_1,..,f_{i-1},f_{i+1},..,f_s) \neq I$.

A minimal standard basis is {\bf completely reduced} if $\forall i: {\tt reduce}(f_i,(f_1,..,f_{i-1},f_{i+1},..,f_s))=f_i$
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A function $\hbox{NF} : R^r \times \{G \mid G\ \hbox{ a standard
basis}\} \to R^r, (p,G) \mapsto \hbox{NF}(p|G)$, is called a {\bf normal
form} if for any $p \in R^r$ and any standard basis $G$ the following
holds: if $\hbox{NF}(p|G) \not= 0$ then $L(g)$ does not divide
$L(\hbox{NF}(p|G))$ for all $g \in G$.
The function may also be applied to any generating set of an ideal:
the result is then not uniquely defined.

\noindent
$\hbox{NF}(p|G)$ is called a {\bf normal form of} $p$ {\bf with
respect to} $G$
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For a standard basis $G$ of $I$ the following holds:
$f \in I$ if and only if $\hbox{NF}(f,G) = 0$.
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Let \hbox{$I \subseteq K[\underline{x}]^r$} be a homogeneous module, then the Hilbert function
$H_I$ of $I$ (see below)
and the Hilbert function $H_{L(I)}$ of the leading module $L(I)$
coincide, i.e.,
$H_I=H_{L(I)}$.
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Let M $=\bigoplus_{i\in Z} M_i$ be a graded module over $K[x_1,..,x_n]$ with
respect to weights $(w_1,..w_n)$.
The {\bf Hilbert function} of $M$, $H_M$, is defined (on the integers) by
$$H_M(k) :=dim_K M_k.$$
The {\bf Hilbert-Poincare series}  of $M$ is the power series
$$\hbox{HP}_M(t) :=\sum_{i=-\infty}^\infty
H_M(i)t^i=\sum_{i=-\infty}^\infty dim_K M_i \cdot t^i.$$
It turns out that $\hbox{HP}_M(t)$ can be written in two useful ways
for weights $(1,..,1)$:
$$\hbox{HP}_M(t)={Q(t)\over (1-t)^n}={P(t)\over (1-t)^{dim(M)}}$$
where $Q(t)$ and $P(t)$ are polynomials in ${\bf Z}[t]$.
$Q(t)$ is called the {\bf first Hilbert series},
and $P(t)$ the {\bf second Hilbert series}.
If \hbox{$P(t)=\sum_{k=0}^N a_k t^k$}, and \hbox{$d = dim(M)$},
then \hbox{$H_M(s)=\sum_{k=0}^N a_k$ ${d+s-k-1}\choose{d-1}$}
(the {\bf Hilbert polynomial}) for $s \ge N$.
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Generalizing this to quasihomogeneous modules we get
$$\hbox{HP}_M(t)={Q(t)\over {\Pi_{i=1}^n(1-t^{w_i})}}$$
where $Q(t)$ is a polynomial in ${\bf Z}[t]$.
$Q(t)$ is called the {\bf first (weighted) Hilbert series} of M.
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Let $R$ be a quotient of $\hbox{Loc}_< K[\underline{x}]$ and let \hbox{$I=(g_1, ..., g_s)$} be a submodule of $R^r$.
Then the {\bf module of syzygies} (or {\bf 1st syzygy module}, {\bf module of relations}) of $I$, syz($I$), is defined to be the kernel of the map \hbox{$R^s \rightarrow R^r,\; \sum_{i=1}^s w_ie_i \mapsto \sum_{i=1}^s w_ig_i$.}
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$(k-1)$-st
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Note, that the syzygy modules of $I$ depend on a choice of generators $g_1, ..., g_s$.
But one can show that they depend on $I$ uniquely up to direct summands.
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Let $I=(g_1,...,g_s)\subseteq R^r$ and $M= R^r/I$.
A {\bf free resolution of $M$} is a long exact sequence
$$...\longrightarrow F_2 \buildrel{A_2}\over{\longrightarrow} F_1
\buildrel{A_1}\over{\longrightarrow} F_0\longrightarrow M\longrightarrow
0,$$
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$R=\hbox{Loc}_< K[\underline{x}]$
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Let $R$ be a graded ring (e.g., $R = \hbox{Loc}_< K[\underline{x}]$) and
let $I \subset R^r$ be a graded submodule. Let
$$
  R^r = \bigoplus_a R\cdot e_{a,0} \buildrel A_1 \over \longleftarrow
        \bigoplus_a R\cdot e_{a,1} \longleftarrow \ldots \longleftarrow
        \bigoplus_a R\cdot e_{a,n} \longleftarrow 0
$$
be a minimal free resolution of $R^r/I$ considered with homogeneous maps
of degree 0. Then the {\bf graded Betti number} $b_{i,j}$ of $R^r/I$ is
the minimal number of generators $e_{a,j}$ in degree $i+j$ of the $j$-th
syzygy module of $R^r/I$ (i.e., the $(j-1)$-st syzygy module of
$I$). Note that, by definition, the $0$-th syzygy module of $R^r/I$ is $R^r$
and the 1st syzygy module of $R^r/I$ is $I$.
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$s$
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$$
    \hbox{deg}(e_{a,j}) \le s+j-1 \quad \hbox{for all $j$.}
$$
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Let $<$ be the lexicographical ordering on $R=K[x_1,...,x_n]$ with $x_1
< ... < x_n$.
For $f \in R$ let lvar($f$) (the leading variable of $f$) be the largest
variable in $f$,
i.e., if $f=a_s(x_1,...,x_{k-1})x_k^s+...+a_0(x_1,...,x_{k-1})$ for some
$k \leq n$ then lvar$(f)=x_k$.

Moreover, let
\hbox{ini}$(f):=a_s(x_1,...,x_{k-1})$. The pseudoremainder
$r=\hbox{prem}(g,f)$ of $g$ with respect to $f$ is
defined by the equality $\hbox{ini}(f)^a\cdot g = qf+r$ with
$\hbox{deg}_{lvar(f)}(r)<\hbox{deg}_{lvar(f)}(f)$ and $a$
minimal.

A set $T=\{f_1,...,f_r\} \subset R$ is called triangular if
$\hbox{lvar}(f_1)<...<\hbox{lvar}(f_r)$. Moreover, let $ U \subset T $,
then $(T,U)$ is called a triangular system, if $T$ is a triangular set
such that $\hbox{ini}(T)$ does not vanish on $V(T) \setminus V(U)
(=:V(T\setminus U))$.

$T$ is called irreducible if for every $i$ there are no
$d_i$,$f_i'$,$f_i''$ such that
$$   \hbox{lvar}(d_i)<\hbox{lvar}(f_i) =
\hbox{lvar}(f_i')=\hbox{lvar}(f_i''),$$
$$   0 \not\in \hbox{prem}(\{ d_i, \hbox{ini}(f_i'),
\hbox{ini}(f_i'')\},\{ f_1,...,f_{i-1}\}),$$
$$\hbox{prem}(d_if_i-f_i'f_i'',\{f_1,...,f_{i-1}\})=0.$$
Furthermore, $(T,U)$ is called irreducible if $T$ is irreducible.

The main result on triangular sets is the following: Let
$G=\{g_1,...,g_s\} \subset R$, then there are irreducible triangular sets $T_1,...,T_l$
such that $V(G)=\bigcup_{i=1}^{l}(V(T_i\setminus I_i))$
where $I_i=\{\hbox{ini}(f) \mid f \in T_i \}$. Such a set
$\{T_1,...,T_l\}$ is called an {\bf irreducible characteristic series} of
the ideal $(G)$.
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Let $f\colon(C^{n+1},0)\rightarrow(C,0)$ be a complex isolated hypersurface singularity given by a polynomial with algebraic coefficients which we also denote by $f$.
Let $O=C[x_0,\ldots,x_n]_{(x_0,\ldots,x_n)}$ be the local ring at the origin and $J_f$ the Jacobian ideal of $f$.

A {\bf Milnor representative} of $f$ defines a differentiable fibre bundle over the punctured disc with fibres of homotopy type of $\mu$ $n$-spheres.
The $n$-th cohomology bundle is a flat vector bundle of dimension $n$ and carries a natural flat connection with covariant derivative $\partial_t$.
The {\bf monodromy operator} is the action of a positively oriented generator of the fundamental group of the punctured disc on the Milnor fibre.
Sections in the cohomology bundle of {\bf moderate growth} at $0$ form a regular $D=C\{t\}[\partial_t]$-module $G$, the {\bf Gauss-Manin connection}.

By integrating along flat multivalued families of cycles, one can consider fibrewise global holomorphic differential forms as elements of $G$.
This factors through an inclusion of the {\bf Brieskorn lattice} $H'':=\Omega^{n+1}_{C^{n+1},0}/df\wedge d\Omega^{n-1}_{C^{n+1},0}$ in $G$.

The $D$-module structure defines the {\bf V-filtration} $V$ on $G$ by $V^\alpha:=\sum_{\beta\ge\alpha}C\{t\}ker(t\partial_t-\beta)^{n+1}$.
The Brieskorn lattice defines the {\bf Hodge filtration} $F$ on $G$ by $F_k=\partial_t^kH''$ which comes from the {\bf mixed Hodge structure} on the Milnor fibre.
Note that $F_{-1}=H'$.

The induced V-filtration on the Brieskorn lattice determines the {\bf singularity spectrum} $Sp$ by $Sp(\alpha):=\dim_CGr_V^\alpha Gr^F_0G$.
The spectrum consists of $\mu$ rational numbers $\alpha_1,\dots,\alpha_\mu$ such that $e^{2\pi i\alpha_1},\dots,e^{2\pi i\alpha_\mu}$ are the eigenvalues of the monodromy.
These {\bf spectral numbers} lie in the open interval $(-1,n)$, symmetric about the midpoint $(n-1)/2$.

The spectrum is constant under $\mu$-constant deformations and has the following semicontinuity property:
The number of spectral numbers in an interval $(a,a+1]$ of all singularities of a small deformation of $f$ is greater than or equal to that of f in this interval.
For semiquasihomogeneous singularities, this also holds for intervals of the form $(a,a+1)$.

Two given isolated singularities $f$ and $g$ determine two spectra and from these spectra we get an integer.
This integer is the maximal positive integer $k$ such that the semicontinuity holds for the spectrum of $f$ and $k$ times the spectrum of $g$.
These numbers give bounds for the maximal number of isolated singularities of a specific type on a hypersurface $X\subset{P}^n$ of degree $d$:
such a hypersurface has a smooth hyperplane section, and the complement is a small deformation of a cone over this hyperplane section.
The cone itself being a $\mu$-constant deformation of $x_0^d+\dots+x_n^d=0$, the singularities are bounded by the spectrum of $x_0^d+\dots+x_n^d$.

Using the library {\tt gmssing.lib} one can compute the {\bf monodromy}, the V-fitration on $H''/H'$, and the spectrum.
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$f=x^5+x^2y^2+y^5$
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$\exp(2\pi iM)$
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$f$
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$H''/H'$
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$O/J_f\cong H''/H'$
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If the principal part of $f$ is $C$-nondegenerate, one can compute the spectrum using the library {\tt spectrum.lib}.
In this case, the V-filtration on $H''$ coincides with the Newton-filtration on $H''$ which allows to compute the spectrum more efficiently.
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$\tilde{E}_6$ on a surface $X\subset{P}^3$
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$\mu(f)$, the geometric genus $p_g(f)$
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$x^7+y^7+z^7=0$
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${3 \over 7}, {4 \over 7}, {5 \over 7}, {6 \over 7}, {1 \over 1},
{8 \over 7}, {9 \over 7}, {10 \over 7}, {11 \over 7}, {12 \over 7},
{13 \over 7}, {2 \over 1}, {15 \over 7}, {16 \over 7}, {17 \over 7},
{18 \over 7}$
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The singularities of type $\tilde{E}_6$ form a
$\mu$-constant one parameter family given by
$x^3+y^3+z^3+\lambda xyz=0,\quad \lambda^3\neq-27$.
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$x^3+y^3+z^3$.
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$\tilde{E}_6$ on a septic in $P^3$. But $x^7+y^7+z^7$
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$\tilde{E}_6$.
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Let $A$ denote an $m\times n$ matrix with integral coefficients. For $u
\in Z\!\!\! Z^n$, we define $u^+,u^-$ to be the uniquely determined
vectors with nonnegative coefficients and disjoint support (i.e.,
$u_i^+=0$ or $u_i^-=0$ for each component $i$) such that
$u=u^+-u^-$. For $u\geq 0$ component-wise, let $x^u$ denote the monomial
$x_1^{u_1}\cdot\ldots\cdot x_n^{u_n}\in K[x_1,\ldots,x_n]$.

The ideal
$$ I_A:=(x^{u^+}-x^{u^-} | u\in\ker(A)\cap Z\!\!\! Z^n)\ \subset
K[x_1,\ldots,x_n] $$
is called a \bf toric ideal. \rm

The first problem in computing toric ideals is to find a finite
generating set: Let $v_1,\ldots,v_r$ be a lattice basis of $\ker(A)\cap
Z\!\!\! Z^n$ (i.e, a basis of the $Z\!\!\! Z$-module). Then
$$ I_A:=I:(x_1\cdot\ldots\cdot x_n)^\infty $$
where
$$ I=<x^{v_i^+}-x^{v_i^-}|i=1,\ldots,r> $$
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section Algorithms.
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computes $I_A$ via the
extended matrix $B=(I_m|A)$,
where $I_m$ is the $m\times m$ unity matrix. A lattice basis of $B$ is
given by the set of vectors $(a^j,-e_j)\in Z\!\!\! Z^{m+n}$, where $a^j$
is the $j$-th row of $A$ and $e_j$ the $j$-th coordinate vector. We
look at the ideal in $K[y_1,\ldots,y_m,x_1,\ldots,x_n]$ corresponding to
these vectors, namely
$$ I_1=<y^{a_j^+}- x_j y^{a_j^-} | j=1,\ldots, n>.$$
We introduce a further variable $t$ and adjoin the binomial $t\cdot
y_1\cdot\ldots\cdot y_m -1$ to the generating set of $I_1$, obtaining
an ideal $I_2$ in the polynomial ring $K[t,
y_1,\ldots,y_m,x_1,\ldots,x_n]$. $I_2$ is saturated w.r.t. all
variables because all variables are invertible modulo $I_2$. Now $I_A$
can be computed from $I_2$ by eliminating the variables
$t,y_1,\ldots,y_m$.
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$v_1,\ldots,v_r$
for the integer kernel of $A$
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$$ I_1=<x^{v_i^+}-x^{v_i^-}|i=1,\ldots,r> $$
is saturated -- as in the algorithm of Conti and Traverso -- by
inversion of all variables: One adds an auxiliary variable $t$ and the
generator $t\cdot x_1\cdot\ldots\cdot x_n -1$ to obtain an ideal $I_2$
in $K[t,x_1,\ldots,x_n]$ from which one computes $I_A$ by elimination of
$t$.
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compute $I_A$ without any auxiliary variables, provided that $A$ contains a vector $w$
with positive coefficients in its row space. This is a real restriction,
i.e., the algorithm will not necessarily work in the general case.

A lattice basis $v_1,\ldots,v_r$ is again computed via the
LLL-algorithm.
The saturation step is performed in the following way:
First note that $w$ induces a positive grading w.r.t. which the ideal
$$ I=<x^{v_i^+}-x^{v_i^-}|i=1,\ldots,r> $$
is homogeneous corresponding to our lattice basis. We use the following
lemma:

Let $I$ be a homogeneous ideal w.r.t. the weighted reverse
lexicographical ordering with weight vector $w$ and variable order $x_1
> x_2 > \ldots > x_n$. Let $G$ denote a Groebner basis of $I$ w.r.t.
this ordering.  Then a Groebner basis of $(I:x_n^\infty)$ is obtained by
dividing each element of $G$ by the highest possible power of $x_n$.

From this fact, we can succesively compute
$$ I_A= I:(x_1\cdot\ldots\cdot x_n)^\infty
=(((I:x_1^\infty):x_2^\infty):\ldots :x_n^\infty); $$
in the $i$-th step we take $x_i$ as the smallest variable and apply the
lemma with $x_i$ instead of $x_n$.

This procedure involves $n$ Groebner basis computations. Actually, this
number can be reduced to at most $n/2$
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to $n/2$ Groebner basis
computations. It needs no auxiliary variables, but a supplementary
precondition; namely, the existence of a vector without zero components
in the kernel of $A$.

The main idea comes from the following observation:

Let $B$ be an integer matrix, $u_1,\ldots,u_r$ a lattice basis of the
integer kernel of $B$. Assume that all components of $u_1$ are
positive. Then
$$ I_B=<x^{u_i^+}-x^{u_i^-}|i=1,\ldots,r>, $$
i.e., the ideal on the right is already saturated w.r.t. all variables.

The algorithm starts by finding a lattice basis $v_1,\ldots,v_r$ of the
kernel of $A$ such that $v_1$ has no zero component. Let
$\{i_1,\ldots,i_l\}$ be the set of indices $i$ with
$v_{1,i}<0$. Multiplying the components $i_1,\ldots,i_l$ of
$v_1,\ldots,v_r$ and the columns $i_1,\ldots,i_l$ of $A$ by $-1$ yields
a matrix $B$ and a lattice basis $u_1,\ldots,u_r$ of the kernel of $B$
that fulfill the assumption of the observation above. It is then possible
to compute a generating set of $I_A$ by applying the following
``variable flip'' successively to $i=i_1,\ldots,i_l$:

Let $>$ be an elimination ordering for $x_i$. Let $A_i$ be the matrix
obtained by multiplying the $i$-th column of $A$ by $-1$. Let
$$\{x_i^{r_j} x^{a_j} - x^{b_j} | j\in J \}$$
be a Groebner basis of $I_{A_i}$ w.r.t. $>$ (where $x_i$ is neither
involved in $x^{a_j}$ nor in $x^{b_j}$). Then
$$\{x^{a_j} - x_i^{r_j} x^{b_j} | j\in J \}$$
is a generating set for $I_A$.
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variable $u$ and one supplementary generator $x_1\cdot\ldots\cdot x_n -
u$ (instead of the generator $t\cdot x_1\cdot\ldots\cdot x_n -1$ in
the algorithm of Pottier). The algorithm uses a quite unusual technique to
get rid of the variable $u$ again.
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Let $A$ be an $m\times n$ matrix with integral coefficients, $b\in
Z\!\!\! Z^m$ and $c\in Z\!\!\! Z^n$. The problem
$$ \min\{c^T x | x\in Z\!\!\! Z^n, Ax=b, x\geq 0\hbox{
component-wise}\} $$
is called an instance of the \bf integer programming problem \rm or
\bf IP problem. \rm

The IP problem is very hard; namely, it is NP-complete.

For the following discussion let $c\geq 0$ (component-wise). We
consider $c$ as a weight vector; because of its nonnegativity, $c$ can
be refined into a monomial ordering $>_c$. It turns out that we can
solve such an IP instance with the help of toric ideals:

First we assume that an initial solution $v$ (i.e., $v\in Z\!\!\!
Z^n, v\geq 0, Av=b$) is already known. We obtain the optimal solution
$v_0$ (i.e., with $c^T v_0$ minimal) by the following procedure:
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$>_c$
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$x^v$
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$x^w$
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  Let $F_q$ be a field with $q$ elements. A \emph{linear code} $C$ is a linear subspace of $F_q^n$ endowed with the
  {\bf Hamming metric}.
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  {\bf Hamming distance} between {\bf x,y}$\in F_q^n: d(x,y)=\#\{i|x_i\ne y_i\}$.
  {\bf Hamming weight} of {\bf x}$\in F_q^n: wt(x)=\#\{i|x_i\ne 0\}$.
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  {\bf Minimum distance} of the code $C: d(C):=\min_{{\bf x,y}\in C, {\bf x}\ne {\bf y}}(d({\bf x,y}))$.
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  The code $C$ of dimension $k$ and minimum distance $d$ is denoted as $[n,k,d]$.
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  A matrix $G$ whose rows are the base vectors of $C$ is the {\bf generator matrix}.
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  A matrix $H$ with the property ${\bf c}\in C\iff H{\bf c}^T=0$ is the {\bf check matrix}.
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  The code $C$ is {\bf cyclic}, if for every codeword ${\bf c}=(c_0,\dots,c_{n-1})$ in $C$ its
  cyclic shift $(c_{n-1},c_0,\dots,c_{n-2})$ is again a codeword in $C$.
  When working with cyclic codes, vectors are usually presented as polynomials.
  So ${\bf c}$ is represented by the polynomial $c(x)=\sum_{i=0}^{n-1}c_ix^i$ with $x^n=1$, more
  precisely $c(x)$ is an element of the factor ring $F_q[X]/\langle X^n-1 \rangle$.
  Cyclic codes over $F_q$ of length $n$ correspond one-to-one to ideals in this factor ring.
  We assume for cyclic codes that $(q,n)=1$. Let $F=F_{q^m}$ be
  the splitting field of $X^n-1$ over $F_q$. Then $F$ has a {\bf primitive $n$-th root of unity} which will be denoted by $a$.
  A cyclic code is uniquely given by a {\bf defining set} $S_C$ which is a subset of $Z\!\!\! Z_n$ such that
  $$
    c(x) \in C \ \hbox{ if } \ c(a ^i )=0 \ \hbox{ for all } \ i \in S_C.
  $$
  A cyclic code has several defining sets.
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  {\bf Complete decoding}: Given $y\in F_q^n$ and a code $C\subseteq F_q^n$, so that $y$ is at distance $d(y,C)$ from
  the code, find $c\in C: d(y,c)=d(y,C)$.
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  {\bf Bounded up to half the minimum distance}: With the additional assumption $d({\bf y},C)\le(d(C)-1)/2$, a codeword with the above property
  is unique.
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  {\bf Generic decoding}: Solve some system $S(C)$ and obtain some "closed" formulas $CF$. Evaluating these formulas
  at data specific to a received word ${\bf r}$ should yield a solution to the decoding problem. For example for
  $f\in CF: f(syndrome({\bf r}),x)=poly(x)$. The roots of $poly(x)=0$ yield error positions, see the section on the
  general error-locator polynomial.
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  {\bf Online decoding}: Solve some system $S(C,{\bf r})$. The solutions should solve the decoding problem.
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  Let $C$ be an $[n,k]$ cyclic code over $F_q$; $F$ is a splitting field with $a$ being a primitive n-th root of unity. Let
  $S_C=\{i_1,\dots,i_{n-k}\}$ be the complete defining set of $C$. Let ${\bf r}={\bf c}+{\bf e}$
  be a received word with ${\bf c}\in C$ and ${\bf e}$ an error vector.
  Denote the corresponding polynomials in $F_q[x]/\langle x^n-1 \rangle$ by $r(x)$, $c(x)$ and $e(x)$, resp. Compute syndromes
  $$
    s_{i_m}=r(a^{i_m})=e(a^{i_m})= \sum_{l=1}^{t}e_{j_l}(a^{i_m})^{j_l}, \ \ 1\le m \le n-k,
  $$
  where $t$ is the number of errors, $j_1,\dots,j_t$ are the {\bf error positions} and $e_{j_1},\dots,e_{j_t}$ are the {\bf
  error values}.
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  Define $z_l=a^{j_l}$ and $y_l=e_{j_l}$. Then $z_1,\dots,z_t$ are the error locations and $y_1,\dots,y_t$ are the error values and
  the syndromes above become {\bf generalized power sum functions} $s_{i_m}=\sum_{l=1}^{t}y_lz_l^{i_m}, \ \ 1\le m \le n-k. $
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<!-- l2h_begin sing_l2h 573 -->
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  $f_u := \sum_{l=1}^e Y_lZ_l^{i_u}-X_u=0, 1\le u\le n-k$;
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  $\epsilon_j:=X_j^{q^m}-X_j=0, 1\le j\le n-k,$ since $s_j\in F$;
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\end{rawhtml}
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<!-- l2h_begin sing_l2h 575 -->
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  $\eta_i:= Z_i^{n+1}-Z_i=0, 1\le i\le e$, since $a^{j_i}$ are either $n$-th roots of unity or zero;
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  $\lambda_i:=Y_i^{q-1}-1=0, 1\le i\le e$, since $y_l\in F_q \setminus\{0\}$.
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\end{rawhtml}
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<!-- l2h_begin sing_l2h 577 -->
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  We obtain the following set of polynomials in the variables $X=(X_1,\dots,X_{n-k})$, $Z=(Z_1,\dots,Z_e)$  and  $Y=(Y_1,\dots,Y_e)$:
  $$ F_C=\{f_j,\epsilon_j,\eta_i,\lambda_i:1\le j\le n-k,1\le i\le e\}\subset F_q[X,Z,Y].$$
  The zero-dimensional ideal $I_C$ generated by $F_C$ is the {\bf CRHT-syndrome ideal}
  associated to the code $C$, and the variety $V(F_C)$ defined by $F_C$ is the {\bf CRHT-syndrome variety},
  after Chen, Reed, Helleseth and Truong.
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  Adding some more polynomials to $F_C$, thus obtaining some $F'_C$, it is possible to prove the following {\bf Theorem}:
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<!-- l2h_begin sing_l2h 579 -->
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  Every cyclic code $C$ possesses a {\bf general error-locator polynomial} $L_C$ from $F_q[X_1,\ldots ,X_{n-k},Z]$ that satisfies the following
  two properties:
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<!-- l2h_begin sing_l2h 580 -->
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  $L_C=Z^e+a_{t-1}Z^{e-1}+\dots +a_0$  with $a_j\in F_q[X_1,\dots ,X_{n-k}],\ 0\le j\le e-1$, where $e$ is the error-correcting
  capacity;
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<!-- l2h_begin sing_l2h 581 -->
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    given a syndrome ${\bf s}=(s_{i_1},\dots,s_{i_{n-k}})\in F^{n-k}$ corresponding to an error of weight $t\le e$
    and error locations $\{k_1,\dots,k_{t}\}$, if we evaluate the $X_u=s_{i_u}$ for all $1\le u \le n-k$,
    then the roots of $L_C({\bf s},Z)$ are exactly $a^{k_1},\dots,a^{k_t}$ and 0 of multiplicity $e-t$, in other words
    $L_C({\bf s},Z)=Z^{e-t}\prod_{i=1}^{t}(Z-a ^{k_i}).$
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  The general error-locator polynomial  actually is an element of the reduced Gr\"obner basis of $\langle F'_C \rangle$. Having this polynomial,
  decoding of the cyclic code $C$ reduces to univariate factorization.
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<!-- l2h_begin sing_l2h 583 -->
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  Let $C$ be the binary $[n,k,d]$ cyclic code with the defining set $S_C=\{i_1,\dots,i_v\}$. Let $1\le w\le n$ and let $J_C(w)$ denote the system:
  $$
  Z_1^{i_1}+\dots+Z_w^{i_1}=0,
  $$
  $$
  \vdots
  $$
  $$
  Z_1^{i_v}+\dots+Z_w^{i_v}=0,
  $$
  $$
  Z_1^n-1=0,
  $$
  $$
  \vdots
  $$
  $$
  Z_w^n-1=0,
  $$
  $$
  p(n,Z_i,Z_j)=0, 1\le i<j\le w.
  $$
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<!-- l2h_begin sing_l2h 584 -->
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  Then the number of solutions of $J_C(w)$ is equal to $w!$ times the number of codewords of weight $w$. And for $1\le w\le d$, either $J_C(w)$
  has no solutions, which is equivalent to $w<d$, or $J_C(w)$ has some solutions, which is equivalent to $w=d$.
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<!-- l2h_begin sing_l2h 585 -->
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$p$
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  The {\bf error-locator polynomial} is defined by
  $$
    \sigma(Z)=\prod_{l=1}^{t}(Z-z_l).
  $$
  If this product is expanded,
  $$
    \sigma(Z)=Z^t+\sigma_1Z^{t-1}+\dots+\sigma_{t-1}Z+\sigma_t,
  $$
  then the coefficients $\sigma_i$ are the {\bf elementary symmetric functions} in
  the error locations $z_1,\dots,z_t$
  $$
    \sigma_i=(-1)^i\sum_{1\le j_1<j_2<\dots<j_i\le t}z_{j_1}z_{j_2}\dots z_{j_i}, \ 1\le i\le t.
  $$
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  The syndromes $s_i=r(a^i)=e(a^i)$ and the coefficients $\sigma_i$ satisfy
  the following {\bf generalized Newton identities}:
  $$
    s_i+\sum_{j=1}^{t}\sigma_js_{i-j}=0,\ \  \hbox{ for all } \ i\in Z\!\!\! Z_n.
  $$
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<!-- l2h_begin sing_l2h 588 -->
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  We have $ s_{i+n}=s_i, \hbox{ for all }  i\in Z\!\!\! Z_n$, since $s_{i+n}=r(a^{i+n})=r(a^i)$. Furthermore
  $$
    s_i^q=(e(a^i))^q=e(a^{iq})=s_{qi}, \hbox{ for all } i \in Z_n,
  $$
  and $\sigma_i^{q^m}=\sigma_i, \hbox{ for all } 1\le i \le t$.
  Replace the syndromes by variables and obtain the following set of polynomials $Newton_t$ in the variables
  $S_1,\dots,S_n$ and $\sigma_1,\dots,\sigma_t$:
  $$
  \sigma_i^{q^m}-\sigma_i, \ \  \forall \ 1\le i \le t,
  $$
  $$
  S_{i+n}-S_i, \ \  \forall \  i\in Z\!\!\! Z_n,
  $$
  $$
  S_i^q-S_{qi}, \ \  \forall \  i\in Z\!\!\! Z_n,
  $$
  $$
  S_i+\sum_{j=1}^{t}\sigma_jS_{i-j}, \ \  \forall \  i\in Z\!\!\! Z_n,
  $$
  $$
  S_i-s_i(r) \ \ \forall \ i\in S_C.
  $$
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<!-- l2h_begin sing_l2h 589 -->
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  Let $I=\langle g_1,\dots,g_m \rangle \subseteq F_q[X_1,\dots,X_s]$ be an
  ideal. Define
  $$
    I_q:=I+\langle X_1^q-X_1,\dots,X_s^q-X_s\rangle .
  $$
  So $I_q$ is a zero-dimensional ideal. Define also $V(I_q)=:\{P_1,\dots,P_n\}$.
  Every $q$-ary linear code $C$ with parameters $[n,k]$ can be seen as an
  {\bf affine variety code} $C(I,L)$, that is, the image of a vector space $L$ of the {\bf evaluation map}
  $$
  \phi :R\to F_q ^n
  $$
  $$
  \bar{f}\mapsto(f(P_1),\dots,f(P_n)),
  $$
  where $R:=F_q[U_1,\dots,U_s]/I_q$, $L$ is a vector subspace of $R$ and $\bar{f}$ the coset of $f$ in $F_q[U_1,\dots,U_s]$ modulo $I_q$.
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<!-- l2h_begin sing_l2h 590 -->
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  Given a $q$-ary $[n,k]$ code $C$ with a generator matrix $G=(g_{ij})$:
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<!-- l2h_begin sing_l2h 591 -->
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  choose $s$, such that $q^s\ge n$, and construct $s$ distinct points $P_1,\dots,P_s$ in $F_q^s$.
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<!-- l2h_begin sing_l2h 592 -->
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  Construct a Gr\"obner basis $\{g_1,\dots,g_m\}$ for an ideal $I$ of polynomials from $F_q[X_1,\dots,X_s]$ that vanish at the points
  $P_1,\dots,P_s$. Define $\xi_i\in F_q[X_1,\dots,X_s]$ such that $\xi_i(P_i)=1,\xi_i(P_j)=0,i\ne j$.
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  Then $f_i=\sum_{i=1}^ng_{ij}\xi_j$ span the space $L$, so that $g_{ij}=f_i(P_j)$.
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  In this way we obtain that the code $C$ is the image of the evaluation above, thus $C=C(I,L)$. In the
  same way by considering a parity check matrix instead of a generator matrix we have that the dual code is also an affine variety code.

  The method of decoding is a generalization of CRHT. One needs to add polynomials
  $(g_l(X_{k1},\dots,X_{ks}))_{l=1,\dots,m; k=1,\dots,t}$ for every error position. We also assume that field equations on $X_{ij}$'s are included
  among the polynomials above. Let $C$ be a $q$-ary $[n,k]$ linear code such that its
  dual is written as an affine variety code of the form $C^{\bot}=C(I,L)$.
  Let ${\bf r}={\bf c}+{\bf e}$ as usual and $t\le e$. Then the syndromes are computed by $s_i=\sum_{j=1}^nr_jf_i(P_j)=\sum_{j=1}^ne_jf_i(P_j)
  \hbox{ for } i=1,\dots,n-k$.
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<!-- l2h_begin sing_l2h 595 -->
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  Consider the ring $F_q[X_{11},\dots,X_{1s},\ldots,X_{t1},\dots,X_{ts},E_1,\dots,E_t]$, where $(X_{i1},\dots,X_{is})$ correspond to
  the $i$-th error position and $E_i$ to the $i$-th error value. Consider the ideal $Id_C$ generated by
  $$
  \sum_{j=1}^tE_jf_i(X_{j1},\dots,X_{js})-s_i, 1\le i\le n-k,
  $$
  $$
  g_l(X_{j1},\dots,X_{js}), 1\le l\le m,
  $$
  $$
  E_k^{q-1}-1.
  $$
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  Let $G$ be the reduced Gr\"obner basis for $Id_C$ with respect to an elimination order $X_{11}<\dots<X_{1s}<E_1$.
  Then we may solve for the error locations and values by applying elimination theory to the polynomials in $G$.
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<!-- l2h_begin sing_l2h 597 -->
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  Let ${\bf b}_1,\dots,{\bf b}_n$ be a basis of $F_q^n$ and let $B$ be the $n \times n$ matrix with
  ${\bf b}_1,\dots,{\bf b}_n$ as rows. The {\bf unknown syndrome} ${\bf u}(B,{\bf e})$ of a word ${\bf e}$ w.r.t $B$
  is the column vector ${\bf u}(B,{\bf e})=B{\bf e}^T$ with entries $u_i(B,{\bf e})={\bf b}_i\cdot {\bf e}$ for $i=1,\dots,n$.

  For two vectors ${\bf x,y} \in F_q^n$ define
  ${\bf x*y}=(x_1y_1,\dots,x_ny_n)$. Then ${\bf b}_i*{\bf b_j}$ is a linear combination of
  ${\bf b}_1,\dots,{\bf b}_n$, so there are constants $\mu^{ij}_l\in F_q $ such that
  ${\bf b}_i*{\bf b}_j = \sum_{l=1}^n \mu^{ij}_l{\bf b}_l.$
  The elements $\mu^{ij}_l\in F_q$ are the {\bf structure constants} of the basis ${\bf b}_1,\dots,{\bf b}_n$.

  Let $B_s$ be the $s \times n$ matrix with ${\bf b}_1,\dots,{\bf b}_s$ as rows ($B=B_n$).
  Then ${\bf b}_1,\dots,{\bf b}_n$ is an {\bf ordered MDS basis} and $B$ an {\bf MDS matrix} if all
  the $s\times s$ submatrices of $B_s$ have rank $s$ for all $s=1 ,\dots ,n$.
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  Let $C$ be an $F_q$-linear code with parameters $[n,k,d]$. W.l.o.g $n\le q$. $H$ is a check matrix of $C$.
  Let ${\bf h}_1,\dots,{\bf h}_{n-k}$ be the rows of $H$.
  One can express ${\bf h}_i = \sum _{j=1}^n a_{ij}{\bf b}_j$ with some $a_{ij}\in F_q$.
  In other words $H=AB$ where $A$ is the $(n-k) \times n$ matrix with entries $a_{ij}$.

  Let ${\bf r}={\bf c}+{\bf e}$ be a received word with ${\bf c}\in C$ and ${\bf e}$ an error vector.
  The syndromes of ${\bf r}$ and ${\bf e}$ w.r.t $H$ are equal and known:
  $$
    s_i({\bf r}):={\bf h}_i \cdot {\bf r}={\bf h}_i\cdot {\bf e}=s_i({\bf e}).
  $$
  They can be expressed in the unknown syndromes of ${\bf e}$ w.r.t $B$:
  $$
    s_i({\bf r})=s_i({\bf e}) =\sum _{j=1}^n a_{ij}u_j({\bf e})
  $$
  since ${\bf h}_i = \sum _{j=1}^n a_{ij}{\bf b}_j$ and ${\bf b}_j \cdot {\bf e}=u_j({\bf e})$.
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<!-- l2h_begin sing_l2h 599 -->
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  Let $B$ be an MDS matrix with structure constants $\mu^{ij}_l$.
  Define $U_{ij}$ in the variables $U_1, \dots ,U_n$ by
  $$
    U_{ij}=\sum _{l=1}^n \mu^{ij}_lU_l.
  $$
  The ideal $J({\bf r})$ in $F_q[U_1,\dots,U_n]$ is generated by
  $$
     \sum_{l=1}^n a_{jl}U_l-s_j({\bf r})  \hbox{ for }   j=1,\dots,n-k.
  $$
  The ideal $I(t,U,V)$ in $F_q[U_1,\dots,U_n,V_1,\dots,V_t]$
  is generated by
  $$
    \sum_{j=1}^t U_{ij}V_j-U_{i,t+1} \hbox{ for } i=1,\dots,n
  $$
  Let $J(t,{\bf r})$ be the ideal in $F_q[U_1,\dots,U_n,V_1,\dots,V_t]$
  generated by $J({\bf r})$ and $I(t,U,V)$.
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  Let $B$ be an MDS matrix with structure constants $\mu^{ij}_l$.  Let $H$ be a check matrix of the code $C$ such that $H=AB$ as above.
  Let ${\bf r}={\bf c}+{\bf e}$ be a received word with ${\bf c} \in C$ the codeword sent
  and ${\bf e}$ the error vector. Suppose that  $wt({\bf e})\ne 0$ and $wt({\bf e})\le\lfloor (d(C)-1)/2\rfloor$.
  Let $t$ be the smallest positive integer such that $J(t,{\bf r})$ has a solution $({\bf u,v})$ over the algebraic closure of
  $F_q$. Then
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  $wt({\bf e})=t$ and the solution is unique and of multiplicity one satisfying ${\bf u}={\bf u}({\bf e})$.
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  the reduced Gr\"{o}bner basis $G$ for the ideal $J(t,{\bf r})$ w.r.t any
  monomial ordering is
  $$
  U_i-u_i({\bf e}), i=1,\dots,n,
  $$
  $$
  V_j-v_j, j=1,\dots,t,
  $$
  where $({\bf u}({\bf e}),{\bf v})$ is the unique solution.
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Faug\`ere,
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<!-- l2h_begin sing_l2h 604 -->
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$g \circ h$
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\end{rawhtml}
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<!-- l2h_begin sing_l2h 605 -->
\end{rawhtml}
$\deg(g), \deg(h)>1$
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     $g\circ h  = g(x/c -d) \circ c(h(x)+d)$.
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\end{rawhtml}
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<!-- l2h_begin sing_l2h 607 -->
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$\deg(g) < \deg(f)$
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$\deg(h)\geq 2$
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$\deg(g)$
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$(g_1,\dots ,g_m,h)$
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\end{rawhtml}
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<!-- l2h_begin sing_l2h 611 -->
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$g_i$
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\end{rawhtml}
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<!-- l2h_begin sing_l2h 612 -->
\end{rawhtml}
$f=g_1\circ \dots \circ g_m\circ h$
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\end{rawhtml}
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<!-- l2h_begin sing_l2h 613 -->
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$\deg(f)$
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\end{rawhtml}
$h$
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\end{rawhtml}
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\end{rawhtml}
$f = (x+y)^2+ 2(x+y) +1$
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\end{rawhtml}
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<!-- l2h_begin sing_l2h 616 -->
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$g = x^2+2x+1$
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\end{rawhtml}
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$h = x+y$
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$f:k^n \to k, n\geq 2$
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\end{rawhtml}
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D. Kozen, S. Landau: Polynomial Decomposition Algorithms, \par
  \quad \qquad          J. Symb. Comp. (1989), 7, 445-456. \par
J. von zu Gathen: Functional Decomposition of Polynomials: the Tame Case,\par
  \quad \qquad          J. Symb. Comp. (1990), 9, 281-299. \par
 J. von zur Gathen, J. Gerhard:  Modern computer algebra,  \par
  \quad \qquad          Cambridge University Press, Cambridge, 2003.
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$(g_1,\ldots,g_l,h)$
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\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 621 -->
\end{rawhtml}
$g_1$
\begin{rawhtml}
<!-- l2h_end sing_l2h 621 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 622 -->
\end{rawhtml}
$g_2,\ldots,g_l,h$
\begin{rawhtml}
<!-- l2h_end sing_l2h 622 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 623 -->
\end{rawhtml}
$f=g\circ h, (g_1,\ldots,g_m,h)$
\begin{rawhtml}
<!-- l2h_end sing_l2h 623 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 624 -->
\end{rawhtml}
$C_0 = c,\ C_1 = x,\ C_n = 2/c\cdot x\cdot C_{n-1}-C_{n-2},\ n \geq 2,c\neq 0$.
\begin{rawhtml}
<!-- l2h_end sing_l2h 624 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 625 -->
\end{rawhtml}
$C_m \circ C_n = C_n\circ C_m$
\begin{rawhtml}
<!-- l2h_end sing_l2h 625 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 626 -->
\end{rawhtml}
$T_n$
\begin{rawhtml}
<!-- l2h_end sing_l2h 626 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 627 -->
\end{rawhtml}
$T_n(x)  = \cos(n \cdot \arccos(x))$
\begin{rawhtml}
<!-- l2h_end sing_l2h 627 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 628 -->
\end{rawhtml}
$P_n$
\begin{rawhtml}
<!-- l2h_end sing_l2h 628 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 629 -->
\end{rawhtml}
$P_n(x+ 1/x) = x^n+ 1/x^n$
\begin{rawhtml}
<!-- l2h_end sing_l2h 629 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 630 -->
\end{rawhtml}
$P_n =$
\begin{rawhtml}
<!-- l2h_end sing_l2h 630 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 631 -->
\end{rawhtml}
$T_n=$
\begin{rawhtml}
<!-- l2h_end sing_l2h 631 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 632 -->
\end{rawhtml}
$P_n(x) = 2\cdot T_n(x/2)$
\begin{rawhtml}
<!-- l2h_end sing_l2h 632 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 633 -->
\end{rawhtml}
$C_n(c\cdot x) = c\cdot T_n(x)$
\begin{rawhtml}
<!-- l2h_end sing_l2h 633 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 634 -->
\end{rawhtml}
$C_0 = 1, C_1 = x, C_2 = 1, C_3 = x$
\begin{rawhtml}
<!-- l2h_end sing_l2h 634 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 635 -->
\end{rawhtml}
$(p-1)x^p y + xy^p$
\begin{rawhtml}
<!-- l2h_end sing_l2h 635 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 636 -->
\end{rawhtml}
The ideal $I$ is generated by binomials of type $X^a-X^b$ (multiindex notation)
in the surrounding polynomial ring $K[X]=K[X_1,...,X_n]$. The binomials
represent a congruence on the monoid ${Z}^n$ with residue monoid $M$.
Let $N$ be the image of $M$ in gp($M$)/torsion. Then $N$ is universal in the
sense that every homomorphism from $M$ to an affine monoid factors through $N$.
If $I$ is a prime ideal, then $K[N]= K[X]/I$. In general, $K[N]=K[X]/P$ where
$P$ is the unique minimal prime ideal of $I$ generated by binomials of type
$X^a-X^b$.

The function computes the normalization of $K[N]$ and returns a newly created
polynomial ring of the same Krull dimension, whose variables are
$x(1),...,x(n-r)$, where $r$ is the rank of the matrix with rows $a-b$.
(In general there is no canonical choice for such an embedding.)
Inside this polynomial ring there is an ideal $I$ which lists the algebra
generators of the normalization of $K[N]$.
\begin{rawhtml}
<!-- l2h_end sing_l2h 636 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 637 -->
\end{rawhtml}
The ideal $I$ is generated by binomials of type $X^a-X^b$ (multiindex notation)
in the surrounding polynomial ring $K[X]=K[X_1,...,X_n]$. The binomials
represent a congruence on the monoid ${Z}^n$ with residue monoid $M$.
Let $N$ be the image of $M$ in gp($M$)/torsion. Then $N$ is universal in the
sense that every homomorphism from $M$ to an affine monoid factors through $N$.
If $I$ is a prime ideal, then $K[N]= K[X]/I$. In general, $K[N]=K[X]/P$ where
$P$ is the unique minimal prime ideal of $I$ generated by binomials of type
$X^a-X^b$.

The function computes $K[N]$ and returns a newly created
polynomial ring of the same Krull dimension, whose variables are
$x(1),...,x(n-r)$, where $r$ is the rank of the matrix with rows $a-b$.
(In general there is no canonical choice for such an embedding.)
\begin{rawhtml}
<!-- l2h_end sing_l2h 637 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 638 -->
\end{rawhtml}
$R^T$.
\begin{rawhtml}
<!-- l2h_end sing_l2h 638 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 639 -->
\end{rawhtml}
 Let $T = (K^*)^r$ be the $r$-dimensional torus acting on the polynomial ring
 $R = K[X_1 ,\ldots,X_n]$ diagonally. Such an action can be described as
 follows: there are integers $a_{i,j}$, $i=1,\ldots,r$, $j=1,\ldots,n$, such
 that $(\lambda_1,\ldots,\lambda_r)\in T$ acts by the substitution
$$ X_j \mapsto \lambda_1^{a_{1,j}} \cdots \lambda_r^{a_{r,j}}X_j,
   \quad j=1,\ldots,n.$$
In order to compute the ring of invariants $R^T$ one must specify the matrix
$A=(a_{i,j})$.
\begin{rawhtml}
<!-- l2h_end sing_l2h 639 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 640 -->
\end{rawhtml}
This function computes the ring of invariants of a finite abelian group $G$
acting diagonally on the surrounding polynomial ring $K[X_1,...,X_n]$. The
group is the direct product of cyclic groups generated by finitely many
elements $g_1,...,g_w$. The element $g_i$ acts on the indeterminate $X_j$ by
$g_i(X_j)=\lambda_i^{u_{ij}}X_j$ where $\lambda_i$ is a primitive root of
unity of order equal to $ord(g_i)$. The ring of invariants is generated by all
monomials satisfying the system
$u_{i1}a_1+\ldots+u_{in} a_n \equiv 0$ mod ord$(g_i)$, $i=1,\ldots,w$.
The input to the function is the $w\times(n+1)$ matrix $U$ with rows
$u_{i1}\ldots u_{in}$ ord$(gi)$, $i=1,\ldots,w$. The output is a monomial ideal
listing the algebra generators of the subalgebra of invariants
{$R^G=\{f\in R : g_i f = f$ for all $i=1,\ldots,w\}$}.
\begin{rawhtml}
<!-- l2h_end sing_l2h 640 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 641 -->
\end{rawhtml}
This function computes the ring of invariants of a diagonalizable group
$D = T\times G$ where $T$ is a torus and $G$ is a finite abelian group, both
acting diagonally on the polynomial ring $K[X_1,\ldots,X_n]$. The group
actions are specified by the input matrices A and U. The first matrix specifies
the torus action, the second the action of the finite group. See
torusInvariants and finiteDiagInvariants for more detail. The output is a
monomial ideal listing the algebra generators of the subalgebra of invariants.
\begin{rawhtml}
<!-- l2h_end sing_l2h 641 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 642 -->
\end{rawhtml}
$S$
\begin{rawhtml}
<!-- l2h_end sing_l2h 642 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 643 -->
\end{rawhtml}
A discrete monomial valuation $v$ on $R = K[X_1 ,\ldots,X_n]$ is determined by
the values $v(X_j)$ of the indeterminates. This function computes the
subalgebra $S = \{ f \in R : v_i ( f ) \geq 0,\ i = 1,\ldots,r\}$ for several
such valuations $v_i$, $i=1,\ldots,r$. It needs the matrix $V = (v_i(X_j))$ as
its input.
\begin{rawhtml}
<!-- l2h_end sing_l2h 643 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 644 -->
\end{rawhtml}
A discrete monomial valuation $v$ on $R = K[X_1 ,\ldots,X_n]$ is determined by
the values $v(X_j)$ of the indeterminates. This function computes the
subalgebra $S = \{ f \in R : v_i ( f ) \geq 0,\ i = 1,\ldots,r\}$ for several
such valuations $v_i$, $i=1,\ldots,r$. It needs the matrix $V = (v_i(X_j))$ as
its input.

This function simultaneously determines the $S$-submodule
$M = \{ f \in R : v_i(f) \geq w_i ,\ i = 1,\ldots,r\}$ for integers
$w_1,\ldots\,w_r$. (If $w_i \geq 0$ for all $i$, $M$ is an ideal of $S$.)
The numbers $w_i$ form the $(n+1)$th column of the input matrix.
\begin{rawhtml}
<!-- l2h_end sing_l2h 644 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 645 -->
\end{rawhtml}
$H^i(F(d))$
\begin{rawhtml}
<!-- l2h_end sing_l2h 645 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 646 -->
\end{rawhtml}
$ 2*genus-2 < deg(G) < size(D) $
\begin{rawhtml}
<!-- l2h_end sing_l2h 646 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 647 -->
\end{rawhtml}
$\Omega(G-D)$
\begin{rawhtml}
<!-- l2h_end sing_l2h 647 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 648 -->
\end{rawhtml}
$delta$
\begin{rawhtml}
<!-- l2h_end sing_l2h 648 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 649 -->
\end{rawhtml}
$epsilon + genus$
\begin{rawhtml}
<!-- l2h_end sing_l2h 649 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 650 -->
\end{rawhtml}
$epsilon:=[(deg(G)-3*genus+1)/2]$
\begin{rawhtml}
<!-- l2h_end sing_l2h 650 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 651 -->
\end{rawhtml}
$f in R$
\begin{rawhtml}
<!-- l2h_end sing_l2h 651 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 652 -->
\end{rawhtml}
$R_f f^s$
\begin{rawhtml}
<!-- l2h_end sing_l2h 652 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 653 -->
\end{rawhtml}
$partial_t$
\begin{rawhtml}
<!-- l2h_end sing_l2h 653 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 654 -->
\end{rawhtml}
$f^lpha$
\begin{rawhtml}
<!-- l2h_end sing_l2h 654 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 655 -->
\end{rawhtml}
$p+1$
\begin{rawhtml}
<!-- l2h_end sing_l2h 655 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 656 -->
\end{rawhtml}
$\partial_t$
\begin{rawhtml}
<!-- l2h_end sing_l2h 656 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 657 -->
\end{rawhtml}
$f^\alpha$
\begin{rawhtml}
<!-- l2h_end sing_l2h 657 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 658 -->
\end{rawhtml}
$f \in R$
\begin{rawhtml}
<!-- l2h_end sing_l2h 658 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 659 -->
\end{rawhtml}
{\sc Singular}
\begin{rawhtml}
<!-- l2h_end sing_l2h 659 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 660 -->
\end{rawhtml}
{\sc Plural}
\begin{rawhtml}
<!-- l2h_end sing_l2h 660 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 661 -->
\end{rawhtml}
{\sc Sca}
\begin{rawhtml}
<!-- l2h_end sing_l2h 661 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 662 -->
\end{rawhtml}
{\sc Letterplace}
\begin{rawhtml}
<!-- l2h_end sing_l2h 662 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 663 -->
\end{rawhtml}
{\sc Singular:SCA}
\begin{rawhtml}
<!-- l2h_end sing_l2h 663 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 664 -->
\end{rawhtml}
{\sc Letterpace}
\begin{rawhtml}
<!-- l2h_end sing_l2h 664 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 665 -->
\end{rawhtml}
{\sc Singular:Letterplace}
\begin{rawhtml}
<!-- l2h_end sing_l2h 665 -->
\end{rawhtml}
\begin{rawhtml}
<!-- l2h_begin sing_l2h 666 -->
\end{rawhtml}
{\sc Singular:Plural}
\begin{rawhtml}
<!-- l2h_end sing_l2h 666 -->
\end{rawhtml}
\end{document}
