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

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 \({\...

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Bibliographic Details
Published in:arXiv.org 2015-07
Main Authors: Zhang, Lizhen, Wang, Haoli, Qu, Yongke
Format: Article
Language:English
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Summary: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 \({\rm U}(\mathcal{S}_R)\) be the group of units of \(\mathcal{S}_R\). The Davenport constant \({\rm D}(\mathcal{S}_R)\) 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 \(I\subsetneq [1,\ell]\) with $$\prod\limits_{i\in I} g_i \equiv \prod\limits_{i=1}^{\ell} g_i\pmod f.$$ In this manuscript, we proved that for the case of \(q=2\), $${\rm D}({\rm U}(\mathcal{S}_R))\leq {\rm D}(\mathcal{S}_R)\leq {\rm D}({\rm U}(\mathcal{S}_R))+\delta_f,$$ where \begin{displaymath} \delta_f=\left\{\begin{array}{ll} 0 & \textrm{if \(\gcd(x*(x+1_{\mathbb{F}_2}),\ f)=1_{\F_{2}}\)}\\ 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{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).
ISSN:2331-8422