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Computation of invariants of finite abelian groups

Authors: Evelyne Hubert and George Labahn
Journal: Math. Comp. 85 (2016), 3029-3050
MSC (2010): Primary 13A50, 15A72; Secondary 12Y05, 13P25, 13P15, 14L30
Published electronically: January 14, 2016
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Abstract: We investigate the computation and applications of rational invariants of the linear action of a finite abelian group in the nonmodular case. By diagonalization, such a group action can be described by integer matrices of orders and exponents. We make use of integer linear algebra to compute a minimal generating set of invariants along with the substitution needed to rewrite any invariant in terms of this generating set. In addition, we show how to construct a minimal generating set that consists only 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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Additional Information

Evelyne Hubert
Affiliation: INRIA Méditerranée, 06902 Sophia Antipolis, France

George Labahn
Affiliation: Cheriton School of Computer Science, University of Waterloo, Waterloo, ON, Canada N2L 3G1

Keywords: Finite groups, rational invariants, matrix normal form, polynomial system reduction, constructive Noether's problem
Received by editor(s): January 24, 2014
Received by editor(s) in revised form: September 4, 2014, and April 27, 2015
Published electronically: January 14, 2016
Article copyright: © Copyright 2016 American Mathematical Society