Singular

Hi,

How I use Singular to substitute a polynomials with a polynomials?
All the functions I know in Singular can only substitute variables with polynomials.

Let me explain in more detailes:
for example assume we have the ring K[x, y, z, t] and the polynomial:
f= y^2+(x*(y+4)-y*(t-2)+2*y-z-zt-4)*x+((y+2)*x+x*y+2*z-2*y^2-2*t-4)*y
and I want to substitute
y^2 by x*y+z*t-4x-y and x*y by x^2+x*z-3t-y-1
in f repetedly until we have f without y^2 and without x*y.

Can we do this substitution by Singular?

You can use reduce if you find an ordered ring in which y^2 and xy are bigger than the other elements, so not x>y>z>t, since x^2>xy. The following works:

> ring r=0,(y,x,z,t),lp;
> ideal i=y2-xy-zt+4x+y,xy-x2-xz+3t+y+1;i;
i[1]=y2-yx+y+4x-zt
i[2]=yx+y-x2-xz+3t+1
> poly f=y2-(x*(y+4)-y*(t-2)+2y-z-zt-4)*x+((y+2)*x+xy+2z-2y^2-2t-4)*y;f;
-4y3+2y2x+y2-yx2+yxt-2yx+2yz-2yt-4y-4x2+xzt+xz+4x
> option(redSB);
> ideal s=std(i);s;
s[1]=x3z+4x3+x2z2-x2zt-3x2t+8x2-8xzt-xz+4x-zt+9t2+3t
s[2]=yz+3yt-x2z-4x2-xz2+xzt-4x+4zt+z
s[3]=yx+y-x2-xz+3t+1
s[4]=y2+2y-x2-xz+4x-zt+3t+1
> reduce(f,s);
12yt2-9yt-33y-3x3-4x2zt-x2z-15x2t+36x2-4xz2t+2xz2+4xzt2-2xzt+25xz-7xt-5x+16zt2+zt-2z-3t2-73t-24
I would have answered sooner but haven't looked at Forum for a while.