Singular Manual: initialMalgrange

| ##### 7.5.5.0. initialMalgrange Procedure from library `dmodapp.lib` (see dmodapp_lib). Usage: initialMalgrange(f,[,a,b,v]); f poly, a,b optional ints, v opt. intvec Return: ring, Weyl algebra induced by basering, extended by two new vars t,Dt Purpose: computes the initial Malgrange ideal of a given polynomial w.r.t. the weight vector (-1,0...,0,1,0,...,0) such that the weight of t is -1 and the weight of Dt is 1. Assume: The basering is commutative and over a field of characteristic 0. Note: Activate the output ring with the `setring` command. The returned ring contains the ideal 'inF', being the initial ideal of the Malgrange ideal of f. Varnames of the basering should not include t and Dt. If a<>0, `std` is used for Groebner basis computations, otherwise, and by default, `slimgb` is used. If b<>0, a matrix ordering is used for Groebner basis computations, otherwise, and by default, a block ordering is used. If a positive weight vector v is given, the weight (d,v[1],...,v[n],1,d+1-v[1],...,d+1-v[n]) is used for homogenization computations, where d denotes the weighted degree of f. Otherwise and by default, v is set to (1,...,1). See Noro (2002). Display: If printlevel=1, progress debug messages will be printed, if printlevel>=2, all the debug messages will be printed. Example:
    LIB "dmodapp.lib";
    ring r = 0,(x,y),dp;
    poly f = x^2+y^3+x*y^2;
    def D = initialMalgrange(f);
    setring D;
    inF;
    ==> inF[1]=x*Dt
    ==> inF[2]=2*x*y*Dx+3*y^2*Dx-y^2*Dy-2*x*Dy
    ==> inF[3]=2*x^2*Dx+x*y*Dx+x*y*Dy+18*t*Dt+9*x*Dx-x*Dy+6*y*Dy+4*x+18
    ==> inF[4]=18*t*Dt^2+6*y*Dt*Dy-y*Dt+27*Dt
    ==> inF[5]=y^2*Dt
    ==> inF[6]=2*t*y*Dt+2*x*y*Dx+2*y^2*Dx-6*t*Dt-3*x*Dx-x*Dy-2*y*Dy+2*y-6
    ==> inF[7]=x*y^2+y^3+x^2
    ==> inF[8]=2*y^3*Dx-2*y^3*Dy-3*y^2*Dx-2*x*y*Dy+y^2*Dy-4*y^2+36*t*Dt+18*x*Dx+1\
       2*y*Dy+36
    setring r;
    intvec v = 3,2;
    def D2 = initialMalgrange(f,1,1,v);
    setring D2;
    inF;
    ==> inF[1]=x*Dt
    ==> inF[2]=2*x*y*Dx+3*y^2*Dx-y^2*Dy-2*x*Dy
    ==> inF[3]=4*x^2*Dx-3*y^2*Dx+2*x*y*Dy+y^2*Dy+36*t*Dt+18*x*Dx+12*y*Dy+8*x+36
    ==> inF[4]=18*t*Dt^2+6*y*Dt*Dy-y*Dt+27*Dt
    ==> inF[5]=y^2*Dt
    ==> inF[6]=2*t*y*Dt-y^2*Dx+y^2*Dy-6*t*Dt-3*x*Dx+x*Dy-2*y*Dy+2*y-6
    ==> inF[7]=x*y^2+y^3+x^2
    ==> inF[8]=2*y^3*Dx-2*y^3*Dy-3*y^2*Dx-2*x*y*Dy+y^2*Dy-4*y^2+36*t*Dt+18*x*Dx+1\
       2*y*Dy+36

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