Invariant polynomials and minimal zero sequences

Abstract

A connection is developed between polynomials invariant under abelian permutation of their variables and minimal zero sequences in a finite abelian group. This connection is exploited to count the number of minimal invariant polynomials for various abelian groups.

The Interplay of Invariant Theory with Multiplicative Ideal Theory and with Arithmetic Combinatorics

“…It was B. Schmid ([73, Section 2]) who first formulated a correspondence between a minimal generating system of F[V ] G and minimal product-one sequences over the character group (see also [24]). The next proposition describes in detail the structural interplay.…”

“…In Subsection 4.3 we specialize to abelian groups whose order is not divisible by the characteristic of F. The Noether number β (G) is the supremum over all finite dimensional G-modules V of the maximal degree of an element in a minimal homogeneous generating system of F[V ] G , and the Davenport constant D(G) is the maximal length of a minimal zero-sum sequence over G. We start with a result on the structural connection between F[V ] G and the monoid of zero-sum sequences over G, that lies behind the equality β (G) = D(G). Clearly, the idea here is well known (as far as we know, it was first used by B. Schmid [73], see also [24]). The benefit of the detailed presentation as given in Proposition 4.7 is twofold.…”

Degree bound for separating invariants of abelian groups

“…which was used in [26], and later in [9] or in [4]. In view of the above connection between the Noether number β(G) and the Davenport constant D(G) it is natural to ask for the meaning of β sep (G) in terms of zero-sum sequences.…”

The Indecomposable Solutions of Linear Congruences

“…For m ≤ 38 the sequence A096337 of OEIS [15] indicates the number of indecomposable solutions of (C m ). In [6] these numbers (+1) are even listed for m up to 60. The paper [4] gives a weak asymptotic lower bound.…”