On the Erdős-Burgess constant of the multiplicative semigroup of a factor ring of \mathbbF_q[x]

2018/05/06 by Wang, Haoli, Hao, Jun, Zhang, Lizhen
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1805.02166

Abstract

Let S be a commutative semigroup endowed with a binary associative operation +. An element e of S is said to be idempotent if e+e=e. The \sl Erdős-Burgess constant of S is defined as the smallest ℓ∈ ℕ∪ \∞\ such that any sequence T of terms from S and of length ℓ contains a nonempty subsequence the sum of whose terms is idempotent. Let q be a prime power, and let \F_q[x] be the polynomial ring over the finite field \F_q. Let R=\F_q[x]\diagup K be a quotient ring of \F_q[x] modulo any ideal K. We gave a sharp lower bound of the Erdős-Burgess constant of the multiplicative semigroup of the ring R, in particular, we determined the Erdős-Burgess constant in the case when K is the power of a prime ideal or a product of pairwise distinct prime ideals in \F_q[x].

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