A problem of Wang on Davenport constant for the multiplicative semigroup of the quotient ring of $\F_2[x]$

Abstract

Let $\F_q[x]$ be the ring of polynomials over the finite field $\F_q$, and let $f$ be a polynomial of $\F_q[x]$. Let $R=\frac{\F_q[x]}{(f)}$ be a quotient ring of $\F_q[x]$ with $0\neq R\neq \F_q[x]$. Let ${\mathcal{S}}_R$ be the multiplicative semigroup of the ring $R$, and let $\mathrm{U}({\mathcal{S}}_R)$ be the group of units of ${\mathcal{S}}_R$. The Davenport constant $\mathrm{D}({\mathcal{S}}_R)$ of the multiplicative semigroup ${\mathcal{S}}_R$ is the least positive integer $\ell$ such that for any $\ell$ polynomials $g_1,g_2,\ldots,g_{\ell}\in \F_q[x]$, there exists a subset $I⊊[1,\ell ]$ with $$\prod _{i\in I}{g}_i\equiv \prod _{i=1}^{\ell }{g}_i(\mathrm{mod}f).$$ In this manuscript, we proved that for the case of $q=2$, $$\mathrm{D}(\mathrm{U}({\mathcal{S}}_R))\le \mathrm{D}({\mathcal{S}}_R)\le \mathrm{D}(\mathrm{U}({\mathcal{S}}_R))+{\delta }_f,$$ where \begin{displaymath} δ_f=\left\{\begin{array}{ll} 0 & \textrm{if $\gcd(x*(x+1_{\mathbb{F}_2}),\ f)=1_{\F_{2}}$}\\ 1 & \textrm{if $\gcd (x\ast (x+1_{{\mathbb{F}}_2}),f)\in \{x,x+1_{{\mathbb{F}}_2}\}$}\\ 2 & \textrm{if $gcd(x\ast (x+1_{{\mathbb{F}}_2}),f)=x\ast (x+1_{{\mathbb{F}}_2})$}\\ \end{array} \right. \end{displaymath} which partially answered an open problem of Wang on Davenport constant for the multiplicative semigroup of $\frac{\F_q[x]}{(f)}$ (G.Q. Wang, \emph{Davenport constant for semigroups II,} Journal of Number Theory, 155 (2015) 124--134).

12 pages

References in corpus (1)

• Additively irreducible sequences in commutative semigroups

Cited by in corpus (1)

• Structure of long idempotent-sum free sequences over finite cyclic semigroups