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Estimating the Number Of Roots of Trinomials over Finite Fields

Abstract : We 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.
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https://hal.inria.fr/hal-01350784
Contributor : Alain Monteil <>
Submitted on : Monday, August 1, 2016 - 4:57:22 PM
Last modification on : Friday, August 2, 2019 - 2:28:01 PM

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  • HAL Id : hal-01350784, version 1
  • ARXIV : 1510.01758

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Zander Kelley, Sean Owen. Estimating the Number Of Roots of Trinomials over Finite Fields. MEGA'2015 (Special Issue), Jun 2015, Trento, Italy. ⟨hal-01350784⟩

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