Estimating the number of roots of trinomials over finite fields

We show that univariate trinomials xn+axs+b∈Fq[x] can have at most δ⌊12+q−1δ⌋ distinct roots in Fq, where δ=gcd⁡(n,s,q−1). We also derive explicit trinomials having q roots in Fq 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(δlog⁡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.