Degree bound for separating invariants of abelian groups

TL;DR

This paper establishes bounds on the degrees of separating invariants for finite abelian groups, showing they are generally smaller than those for generating invariants, with specific characterizations for certain group structures.

Contribution

It provides a new bound for separating invariants of abelian groups and characterizes separating monomial sets via zero-sum sequences.

Findings

Universal degree bound for separating invariants is smaller than for generators, except in specific cases.

Characterization of separating monomials using zero-sum sequences.

Identifies conditions under which bounds coincide for certain abelian groups.

Abstract

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.