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A problem of Wang on Davenport constant for the multiplicative semigroup of the quotient ring of $F_2[x]$ | | Lizhen Zhang
; Haoli Wang
; Yongke Qu
; | | Date: |
12 Jul 2015 | | 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 $0
eq R
eq F_q[x]$. Let $mathcal{S}_R$ be the
multiplicative semigroup of the ring $R$, and let ${
m U}(mathcal{S}_R)$ be
the group of units of $mathcal{S}_R$. The Davenport constant ${
m
D}(mathcal{S}_R)$ 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 $Isubsetneq
[1,ell]$ with $$prodlimits_{iin I} g_i equiv prodlimits_{i=1}^{ell}
g_ipmod f.$$ In this manuscript, we proved that for the case of $q=2$, $${
m
D}({
m U}(mathcal{S}_R))leq {
m D}(mathcal{S}_R)leq {
m D}({
m
U}(mathcal{S}_R))+delta_f,$$ where egin{displaymath}
delta_f=left{egin{array}{ll} 0 & extrm{if $gcd(x*(x+1_{mathbb{F}_2}),
f)=1_{F_{2}}$}\ 1 & extrm{if $gcd(x*(x+1_{mathbb{F}_2}), f)in {x,
x+1_{mathbb{F}_2}}$}\ 2 & extrm{if
$gcd(x*(x+1_{mathbb{F}_2}),f)=x*(x+1_{mathbb{F}_2}) $}\ end{array}
ight.
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). | | Source: | arXiv, 1507.3182 | | Services: | Forum | Review | PDF | Favorites | |
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