Sparse univariate polynomials with many roots over finite fields

被引:4
|
作者
Cheng, Qi [1 ]
Gao, Shuhong [2 ]
Rojas, J. Maurice
Wan, Daqing [3 ]
机构
[1] Univ Oklahoma, Sch Comp Sci, Norman, OK 73019 USA
[2] Clemson Univ, Dept Math Sci, Clemson, SC 29634 USA
[3] Univ Calif Irvine, Dept Math, Irvine, CA 92697 USA
基金
美国国家科学基金会;
关键词
Sparse polynomial; t-nomial; Finite field; Descartes; Coset; Torsion; Chebotarev density; Frobenius; Least prime; DIFFIE-HELLMAN DISTRIBUTIONS; PRIME;
D O I
10.1016/j.ffa.2017.03.006
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
Suppose q is a prime power and f is an element of F-q[x] is a univariate polynomial with exactly t monomial terms and degree < q-1. To establish a finite field analogue of Descartes' Rule, Bi, Cheng, and Rojas (2013) proved an upper bound of 2(q-1) (t-2/t-1) on the number of cosets in F-q*; needed to cover the roots of f in F-q*. Here, we give explicit f with root structure approaching this bound: When q is a perfect (t-1)-st power we give an explicit t-nomial vanishing on q(t-2/t-1) distinct cosets of F-q*. Over prime fields F-p, computational data we provide suggests that it is harder to construct explicit sparse polynomials with many roots. Nevertheless, assuming the Generalized Riemann Hypothesis, we find explicit trinomials having Omega (log p/log log p) distinct roots in F-p. (C) 2017 Elsevier Inc. All rights reserved.
引用
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页码:235 / 246
页数:12
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