Some topics on finite fields

• Autor(a) principal: José Alves Oliveira
• Data de Publicação: 2022
• Tipo de documento: Tese
• Idioma: eng
• Título da fonte: Repositório Institucional da UFMG
• Texto Completo: http://hdl.handle.net/1843/40367; https://orcid.org/0000-0003-3346-3848
• topic: Finite fields; Hypersurfaces; Fermat hypersurfaces; Artin-Schreier hypersurfaces; Elliptic curves; Character sums; Gauss sums; Jacobi sums; Purity of Gauss and Jacobi sums; Rational points; Maximal curves; Perfect fields; Rational functions; Iterated maps; Functional graphs; Dynamics over finite fields; Dynamics of polynomial maps; Linearized polynomails; Rank metric codes; Matemática – Teses; Corpos finitos (Álgebra) -Teses; Hipersuperfícies – Teses; Somas de Gauss – Teses; Curvas algébricas – Teses

In this work, we study some theoretical problems in the theory of finite fields that are of interest for a number of applications, such as in coding theory, cryptography and related areas. In particular, we study the number of rational points on hypersurfaces and present bounds for such numbers and explicit formulas in the cases where certain conditions are satisfied. For some of these hypersurfaces, we also provide conditions for the maximality and minimality of the number of rational points with respect to Weil's bound. Another topic of interest in this thesis is the iteration of maps over fields. For example, we study the functional graph associated to the iteration of polynomial maps over finite fields. We also study the number of solutions of the equation $R^{(n)}(x)=\alpha$ over $\overline{\mathbb{F}}_q$ for a rational function $R$. The last topic in the thesis contains a study of code rank metric codes arising from linearized polynomials over $\mathbb{F}_q$, the so called twisted Gabidulin codes.