Computing the Invariants of Finite Abelian Groups

Evelyne Hubert 1 George Labahn 2
1 AROMATH - AlgebRe, geOmetrie, Modelisation et AlgoriTHmes
CRISAM - Inria Sophia Antipolis - Méditerranée , UoA - University of Athens
2 Symbolic Computation Group
SCG - Symbolic Computation Group
Abstract : We investigate the computation and applications of rational invariants of the linear action of a finite abelian group in the non-modular case. By diagonalization, the group action is accurately described by an integer matrix of exponents. We make use of linear algebra to compute a minimal generating set of invariants and the substitution to rewrite any invariant in terms of this generating set. We show how to compute a minimal generating set that consists of polynomial invariants. As an application, we provide a symmetry reduction scheme for polynomial systems whose solution set is invariant by a finite abelian group action. Finally, we also provide an algorithm to find such symmetries given a polynomial system.
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Mathematics of Computation, American Mathematical Society, 2016, 85 (302), pp.3029-3050. 〈http://www.ams.org/journals/mcom/2016-85-302/S0025-5718-2016-03076-8/〉. 〈10.1090/mcom/3076〉
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Evelyne Hubert, George Labahn. Computing the Invariants of Finite Abelian Groups. Mathematics of Computation, American Mathematical Society, 2016, 85 (302), pp.3029-3050. 〈http://www.ams.org/journals/mcom/2016-85-302/S0025-5718-2016-03076-8/〉. 〈10.1090/mcom/3076〉. 〈hal-00921905v4〉

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