Singular

D.7.1.4 primary_invariants_random

Usage:

primary_invariants_random(G1,G2,...,r[,flags]);
G1,G2,...: <matrices> generating a finite matrix group, r: an <int> where -|r| to |r| is the range of coefficients of the random combinations of bases elements, flags: an optional <intvec> with three entries, if the first one equals 0 (also the default), the programme attempts to compute the Molien series and Reynolds operator, if it equals 1, the programme is told that the Molien series should not be computed, if it equals -1 characteristic 0 is simulated, i.e. the Molien series is computed as if the base field were characteristic 0 (the user must choose a field of large prime characteristic, e.g. 32003) and if the first one is anything else, it means that the characteristic of the base field divides the group order, the second component should give the size of intervals between canceling common factors in the expansion of the Molien series, 0 (the default) means only once after generating all terms, in prime characteristic also a negative number can be given to indicate that common factors should always be canceled when the expansion is simple (the root of the extension field does not occur among the coefficients)

Display:

information about the various stages of the programme if the third flag does not equal 0

Return:

primary invariants (type <matrix>) of the invariant ring and if computable Reynolds operator (type <matrix>) and Molien series (type <matrix>), if the first flag is 1 and we are in the non-modular case then an <intvec> is returned giving some of the degrees where no non-trivial homogeneous invariants can be found

Theory:

Bases of homogeneous invariants are generated successively and random linear combinations are chosen as primary invariants that lower the dimension of the ideal generated by the previously found invariants (see "Generating a Noetherian Normalization of the Invariant Ring of a Finite Group" by Decker, Heydtmann, Schreyer (1998)).

LIB "finvar.lib";
ring R=0,(x,y,z),dp;
matrix A[3][3]=0,1,0,-1,0,0,0,0,-1;
list L=primary_invariants_random(A,1);
print(L[1]);
==> z2,x2+y2,x4+y4-z4