Roots of Trinomials over Prime Fields

The origin of this work was the search for a “Descartes’ rule” for finite fields a nontrivial upper bound for the number of roots of sparse polynomials. In [2], Bi, Cheng, and Rojas establish such an upper bound. Then, in [3], Cheng, Gao, Rojas, and Wan show that the bound is essentially optimal in an infinite number of cases by constructing t-nomials with many roots in Fpt . However, the bound’s optimality remains open in other cases. Here, we look at the roots of trinomials over Fp. Let Z(f) denote the zero-set of f(x) = a1 + a2x e2 + · · ·+ atxt ∈ Fq[x]. At first glance, a nontrivial upper bound for |Z(f)| seems unlikely: consider f(x) = x q−1 2 −1, which always has half of the nonzero elements of Fq as roots. However, a key observation of [2] is that sparse polynomials with many roots have a simple, common characterization: they have large values of δ(f) := gcd(e0, . . . , et, q − 1).