Davenport constant of the multiplicative semigroup of the quotient ring $\frac{\F_p[x]}{\langle f(x)\rangle}$

Abstract

Let $\mathcal{S}$ be a finite commutative semigroup. The Davenport constant of $\mathcal{S}$, denoted $D(\mathcal{S})$, is defined to be the least positive integer $d$ such that every sequence $T$ of elements in $\mathcal{S}$ of length at least $d$ contains a subsequence $T^{\prime }$ with the sum of all terms from $T^{\prime }$ equaling the sum of all terms from $T$. Let $\F_p[x]$ be a polynomial ring in one variable over the prime field $\F_p$, and let $f(x)\in \F_p[x]$. In this paper, we made a study of the Davenport constant of the multiplicative semigroup of the quotient ring $\frac{\F_p[x]}{\langle f(x)\rangle}$. Among other results, we mainly prove that, for any prime $p>2$ and any polynomial $f(x)\in \F_p[x]$ which can be factorized into several pairwise non-associted irreducible polynomials in $\F_p[x]$, then $$D({\mathcal{S}}_{f(x)}^p)=D(U({\mathcal{S}}_{f(x)}^p)),$$ where ${\mathcal{S}}_{f(x)}^p$ denotes the multiplicative semigroup of the quotient ring $\frac{\F_p[x]}{\langle f(x)\rangle}$ and $U({\mathcal{S}}_{f(x)}^p)$ denotes the group of units of the semigroup ${\mathcal{S}}_{f(x)}^p$.

9 pages

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