Sparse univariate polynomials with many roots over finite fields

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.