Davenport constant for semigroups II

TL;DR

This paper investigates the Davenport constant for the multiplicative semigroup of quotient rings of polynomial rings over finite fields, establishing that it equals the Davenport constant of the group of units within those semigroups.

Contribution

It proves that for certain quotient rings of polynomial rings over finite fields, the Davenport constant of the semigroup equals that of its group of units, extending previous results to new algebraic structures.

Findings

Davenport constant of the semigroup equals that of the units group.

Results apply to quotient rings of polynomial rings over finite fields.

Provides a new understanding of zero-sum problems in algebraic semigroups.

Abstract

Let $\mathcal{S}$ be a finite commutative semigroup. The Davenport constant of $\mathcal{S}$, denoted ${\rm D}(\mathcal{S})$, is defined to be the least positive integer $\ell$ such that every sequence $T$ of elements in $\mathcal{S}$ of length at least $\ell$ contains a proper subsequence $T'$ ($T'\neq T$) with the sum of all terms from $T'$ equaling the sum of all terms from $T$. Let $q>2$ be a prime power, and let $\F_q[x]$ be the ring of polynomials over the finite field $\F_q$. Let $R$ be a quotient ring of $\F_q[x]$ with $0\neq R\neq \F_q[x]$. We prove that $${\rm D}(\mathcal{S}_R)={\rm D}(U(\mathcal{S}_R)),$$ where $\mathcal{S}_R$ denotes the multiplicative semigroup of the ring $R$, and $U(\mathcal{S}_R)$ denotes the group of units in $\mathcal{S}_R$.