Computing the Invariants of Finite Abelian Groups

International audienceWe 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

Degree bound for separating invariants of abelian groups

It is proved that the universal degree bound for separating polynomial invariants of a finite abelian group (in non-modular characteristic) is strictly smaller than the universal degree bound for generators of polynomial invariants, unless the goup is cyclic or is the direct product of $r$ even order cyclic groups where the number of two-element direct factors is not less than the integer part of the half of $r$. A characterization of separating sets of monomials is given in terms of zero-sum sequences over abelian groups

Degree bound for separating invariants of abelian groups

It is proved that the universal degree bound for separating polynomial invariants of a finite abelian group (in non-modular characteristic) is typically strictly smaller than the universal degree bound for generators of polynomial invariants. More precisely, these degree bounds can be equal only if the group is cyclic or is the direct sum of r even order cyclic groups where the number of two-element direct summands is not less than the integer part of the half of r. A characterization of separating sets of monomials is given in terms of zero-sum sequences over abelian groups. © 2017 American Mathematical Society