Families of diagonal equations over finite fields: estimates and applications

MARIANA PÉREZ; MELINA PRIVITELLI

Otawa

Seminario; Carleton Finite Fields Seminar; 2022

In this work, we study the set of rational solutions, that is, solutions with coordinates in the finite field $mathbb{F}_q$ of $q$ elements, of certain equations and systems defined by families of diagonal equations with coefficients in $mathbb{F}_q$. In [1] and [2] we obtain explicit estimates and results that guarantee the existence of at least a rational solution of these families, by studying geometric properties of the varieties that define these equations. The results obtained complement those existing in the literature (see [3]). Finally we apply these results to a generalization of Waring´s problem and the distribution of solutions of congruences modulo a prime number.[1] M. Pérez and M. Privitelli. Estimates on the number of rational solutions of variants of diagonal equations over finite fields, Finite Fields and Appl. 68 (2020), 30 pp.[2] M. Pérez and M. Privitelli. On the number of solutions of systems of certain diagonal equations over finite fields. Journal of Number Theory (2021).[3] Gary L. Mullen and D. Panario. Handbook of Finite Fields (1st ed.) . Chapman and Hall/CRC, 2013.