Estimating the number of roots of trinomials over finite fields. (March 2017)

• Record Type: Journal Article
• Authors: Kelley, Zander; Owen, Sean W.
• Is Part Of: Journal of symbolic computation. Volume 79(2017)Part 1
• Page Start: 108
• Page End: 118
• Publication Date: 2017-03
• Subjects: Finite field -- Sparse polynomial -- Trinomial -- Poisson distribution -- Chebotarev density; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2016.08.008
• Languages: English
• ISSNs: 0747-7171

Abstract

Abstract: We show that univariate trinomials x n + a x s + b ∈ F q [ x ] can have at most δ ⌊ 1 2 + q − 1 δ ⌋ distinct roots in F q, where δ = gcd ⁡ ( n, s, q − 1 ) . We also derive explicit trinomials having q roots in F q when q is square and δ = 1, thus showing that our bound is tight for an infinite family of finite fields and trinomials. Furthermore, we present the results of a large-scale computation which suggest that an O ( δ log ⁡ q ) upper bound may be possible for the special case where q is prime. Finally, we give a conjecture (along with some accompanying computational and theoretical support) that, if true, would imply such a bound.