Singular Manual: WLemma

| #### D.2.4.18 WLemma Procedure from library `grobcov.lib` (see grobcov_lib). Usage: WLemma(F,A[,options]); The first argument ideal F in Q[x_1,..,x_n][u_1,..,u_m]; The second argument ideal A in Q[x_1,..,x_n]. Calling sequence: ring R=(0,x_1,..,x_n),(u_1,..,u_m),lp; ideal F=f_1(x_1,..,x_n,u_1,..,u_m),.., f_s(x_1,..,x_n,u_1,..,u_m); ideal A=g_1(u_1,..u_m),..,g_s(u_1,..u_m); list # : Options Calling sequence: WLemma(F,A[,options]); Given the ideal F and ideal A it returns the list (lpp,B,S) were B is the reduced Groebner basis of the specialized F over the segment S, subset of V(A) with top A, determined by Wibmer's Lemma. S is determined in P-representation (or optionally in C-representation). The basis is given by I-regular functions. Options: either ("rep", 0) or ("rep",1) the representation of the resulting segment, by default is 0 =P-representation, (default) but can be set to 1=C-representation. Return: list of [lpp,B,S] = [leading power product, basis,segment], being B the reduced Groebner Basis given by I-regular functions in full representation, of the specialized ideal F on the segment S, subset of V(A) with top A. given in P- or C-representation. It is the result of Wibmer's Lemma. See A. Montes , M. Wibmer, "Groebner Bases for Polynomial Systems with parameters". JSC 45 (2010) 1391-1425.) or the book A. Montes. "The Groebner Cover" (Discussing Parametric Polynomial Systems). Note: The basering R, must be of the form Q[a][x] (a=parameters, x=variables). Example:
    LIB "grobcov.lib";
    if(defined(RE)){kill RE;}
    ring RE=(0,a,b,c,d,e,f),(x,y),lp;
    ideal F=a*x^2+b*x*y+c*y^2,d*x^2+e*x*y+f*y^2;
    ideal A=a*e-b*d;
    WLemma(F,A);
    ==> [1]:
    ==>    _[1]=y2
    ==>    _[2]=x2
    ==> [2]:
    ==>    [1]:
    ==>       _[1]=y2
    ==>    [2]:
    ==>       _[1]=(d)*x2+(e)*xy+(f)*y2
    ==>       _[2]=(a)*x2+(b)*xy+(c)*y2
    ==> [3]:
    ==>    [1]:
    ==>       [1]:
    ==>          _[1]=(ae-bd)
    ==>       [2]:
    ==>          [1]:
    ==>             _[1]=(bf-ce)
    ==>             _[2]=(af-cd)
    ==>             _[3]=(ae-bd)
    ==>          [2]:
    ==>             _[1]=(d)
    ==>             _[2]=(a)
    WLemma(F,A,"rep",1);
    ==> [1]:
    ==>    _[1]=y2
    ==>    _[2]=x2
    ==> [2]:
    ==>    [1]:
    ==>       _[1]=y2
    ==>    [2]:
    ==>       _[1]=(d)*x2+(e)*xy+(f)*y2
    ==>       _[2]=(a)*x2+(b)*xy+(c)*y2
    ==> [3]:
    ==>    [1]:
    ==>       _[1]=(ae-bd)
    ==>    [2]:
    ==>       _[1]=(bdf-cde)
    ==>       _[2]=(af-cd)
    ==>       _[3]=(ae-bd)

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