D.6.1.11 invariant_basis

Procedure from library finvar.lib (see finvar_lib).

Usage:

invariant_basis(g,G1,G2,...);
g: an <int> indicating of which degree (>0) the homogeneous basis should be, G1,G2,...: <matrices> generating a finite matrix group

Returns:

the basis (type <ideal>) of the space of invariants of degree g

Theory:

A general polynomial of degree g is generated and the generators of the matrix group applied. The difference ought to be 0 and this way a system of linear equations is created. It is solved by computing syzygies.

Example:

LIB "finvar.lib";
ring R=0,(x,y,z),dp;
matrix A[3][3]=0,1,0,-1,0,0,0,0,-1;
print(invariant_basis(2,A));
→ x2+y2,
→ z2