Singular

7.5.2.0. pIntersectSyz

Usage:

pIntersectSyz(f, I [,p,s,t]); f poly, I ideal, p,t optial ints, p prime

Return:

vector, coefficient vector of the monic polynomial

Purpose:

compute the intersection of an ideal I with the subalgebra K[f]

Assume:

I is given as Groebner basis.

Note:

If the intersection is zero, this procedure might not terminate.
If p>0 is given, this proc computes the generator of the intersection in
char p first and then only searches for a generator of the obtained
degree in the basering. Otherwise, it searches for all degrees by
computing syzygies.
If s<>0, std is used for Groebner basis computations in char 0,
otherwise, and by default, slimgb is used.
If t<>0 and by default, std is used for Groebner basis
computations in char >0, otherwise, slimgb is used.

Display:

If printlevel=1, progress debug messages will be printed,
if printlevel>=2, all the debug messages will be printed.

LIB "bfun.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
poly s = t*Dt;
pIntersectSyz(s,inF);
==> gen(4)-1/36*gen(2)
int p = prime(20000);
pIntersectSyz(s,inF,p,0,0);
==> gen(4)-1/36*gen(2)