https://hal.inria.fr/hal-01350784Kelley, ZanderZanderKelleyDepartment of Mathematics [Texas A&M University] - Texas A&M University [College Station]Owen, SeanSeanOwenUMBC - University of Maryland [Baltimore County] - University of Maryland SystemEstimating the Number Of Roots of Trinomials over Finite FieldsHAL CCSD2015Maximum likelihood degreeData Singular locusML Discriminant.[MATH.MATH-AG] Mathematics [math]/Algebraic Geometry [math.AG]Monteil, Alain2016-08-01 16:57:222019-08-02 14:28:012016-08-01 16:57:22enConference papers1We show that univariate trinomials $x^n + ax^s + b \in \mathbb{F}_q[x]$ can have at most $\delta \Big\lfloor \frac{1}{2} +\sqrt{\frac{q-1}{\delta}} \Big\rfloor$ distinct roots in $\mathbb{F}_q$, where $\delta = \gcd(n, s, q - 1)$. We also derive explicit trinomials having $\sqrt{q}$ roots in $\mathbb{F}_q$ when $q$ is square and $\delta=1$, 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 $O(\delta \log q)$ 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.