A problem of G. Q. Wang on the Davenport constant of the multiplicative semigroup of quotient rings of $\mathbb {F}_2[x]$
Tom 148 / 2017
Streszczenie
Given a finite commutative semigroup $\mathcal{S}$ (written multiplicatively), denote by ${\rm D}(\mathcal{S})$ the Davenport constant of $\mathcal{S}$, the least positive integer $\ell$ such that for any $x_1,\ldots,x_{\ell}\in \mathcal{S}$ there exists a set $I\subsetneq [1,\ell]$ for which $\prod_{i\in I} x_i=\prod_{i=1}^{\ell} x_i$, the equality being interpreted in the conditional unitization of $\mathcal{S}$ to make sense of the left-hand side also in the case when $I=\emptyset$ and $\mathcal{S}$ is not unitary.
Then, let $R$ be the quotient ring of $\mathbb{F}_2[x]$ by the principal ideal generated by a nonconstant polynomial $f\in \mathbb{F}_2[x]$. Moreover, let $\mathcal{S}_R$ be the multiplicative semigroup of the cosets in $R$, and ${\rm U}(\mathcal{S}_R)$ the group of units of $\mathcal{S}_R$.
We prove that $${\rm D}({\rm U}(\mathcal{S}_R))\leq {\rm D}(\mathcal{S}_R)\leq {\rm D}({\rm U}(\mathcal{S}_R))+\delta_f,$$ where $$ \delta_f=\begin{cases}0 &\textrm{if $\gcd(x*(x+1_{\mathbb{F}_2}), f)=1_{\mathbb F_{2}}$,}\\ 1 & \textrm{if $\gcd(x*(x+1_{\mathbb{F}_2}), f)\in \{x, x+1_{\mathbb{F}_2}\}$,}\\ 2 & \textrm{if $\gcd(x*(x+1_{\mathbb{F}_2}),f)=x*(x+1_{\mathbb{F}_2}) $.}\\ \end{cases} $$ This gives a partial answer to an open problem of G. Q. Wang.
