2015/07/12 by Lizhen Zhang, Zhang, Lizhen, Haoli Wang +3 · 1 citation
Computer Science · Mathematics · #Coding theory and cryptography #Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT) #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.1507.03182
openalex publication_date 2015/07/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let \Fq[x] be the ring of polynomials over the finite field \Fq, and let f be a polynomial of \Fq[x]. Let R=(\Fq[x])/((f)) be a quotient ring of \Fq[x] with 0≠ R≠ \Fq[x]. Let SR be the multiplicative semigroup of the ring R, and let \rm U(SR) be the group of units of SR. The Davenport constant \rm D(SR) of the multiplicative semigroup SR is the least positive integer ℓ such that for any ℓ polynomials g1,g2,…,gℓ∈ \Fq[x], there exists a subset I\subsetneq [1,ℓ] with ∏i∈ I gi ≡ ∏i=1ℓ gi\pmod f. In this manuscript, we proved that for the case of q=2, \rm D(\rm U(SR))≤ \rm D(SR)≤ \rm D(\rm U(SR))+δf, where δf=\0 · amp; \textrmif gcd(x*(x+1_\mathbbF2), f)=1_\F2
1 · amp; \textrmif gcd(x*(x+1_\mathbbF2), f)∈ \x, x+1_\mathbbF2\
2 · amp; \textrmif gcd(x*(x+1_\mathbbF2),f)=x*(x+1_\mathbbF2) . which partially answered an open problem of Wang on Davenport constant for the multiplicative semigroup of (\Fq[x])/((f)) (G.Q. Wang, Davenport constant for semigroups II, Journal of Number Theory, 155 (2015) 124--134).