Top
Back: invariant_ring_random
Forward: primary_invariants_random
FastBack:
FastForward:
Up: finvar_lib
Top: Singular Manual
Contents: Table of Contents
Index: Index
About: About this document

D.7.1.3 primary_invariants

Procedure from library finvar.lib (see finvar_lib).

Usage:
primary_invariants(G1,G2,...[,flags]);
G1,G2,...: <matrices> generating a finite matrix group, 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 occurs not 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>) or ring name (type string) where the Molien series can be found in the char p case; 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 those are chosen as primary invariants that lower the dimension of the ideal generated by the previously found invariants (see paper "Generating a Noetherian Normalization of the Invariant Ring of a Finite Group" by Decker, Heydtmann, Schreyer (1998)).

Example:
 
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(A);
print(L[1]);
==> z2,x2+y2,x2y2


Top Back: invariant_ring_random Forward: primary_invariants_random FastBack: FastForward: Up: finvar_lib Top: Singular Manual Contents: Table of Contents Index: Index About: About this document
            User manual for Singular version 4.3.2, 2023, generated by texi2html.