D.6.1.30 relative_orbit_variety

Procedure from library finvar.lib (see finvar_lib).

Usage:

relative_orbit_variety(I,F,s);
I: an <ideal> invariant under the action of a group, F: a 1xm <matrix> defining the invariant ring of this group, s: a <string> giving a name for a new ring

Return:

a Groebner basis (type <ideal>, named G) for the ideal defining the relative orbit variety with respect to I in the new ring (named s)

Theory:

A Groebner basis of the ideal of algebraic relations of the invariant ring generators is calculated, then one of the basis elements plus the ideal generators. The variables of the original ring are eliminated and the polynomials that are left define the relative orbit variety with respect to I.

Example:

LIB "finvar.lib";
ring R=0,(x,y,z),dp;
matrix F[1][3]=x+y+z,xy+xz+yz,xyz;
ideal I=x2+y2+z2-1,x2y+y2z+z2x-2x-2y-2z,xy2+yz2+zx2-2x-2y-2z;
string newring="E";
relative_orbit_variety(I,F,newring);
print(G);
→ 27*y(3)^6-513*y(3)^4+33849*y(3)^2-784,
→ 1475*y(2)+9*y(3)^4-264*y(3)^2+736,
→ 8260*y(1)+9*y(3)^5-87*y(3)^3+5515*y(3)
basering;
→ //   characteristic : 0
→ //   number of vars : 3
→ //        block   1 : ordering lp
→ //                  : names    y(1) y(2) y(3) 
→ //        block   2 : ordering C