Sparse univariate polynomials with many roots over finite fields

Suppose q is a prime power and f ∈ Fq[x] is a univariate polynomial with exactly t nonzero 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: For q a t power we give an explicit t-nomial vanishing on q t−2 t + · · · + q 1t + 1 distinct cosets of F∗ q . Over prime fields Fp, computational data we provide suggests that it is harder to construct explicit sparse polynomials with many cosets of roots. Nevertheless, we find trinomials vanishing on Ω ( log log p log log log p ) distinct cosets in F∗ p and, assuming the Generalized Riemann Hypothesis, Ω (