Topics in Finite Fields and Combinatorics

• Grant number: 25/02198-0
• Support Opportunities: Scholarships abroad - Research Internship - Post-doctor
• Start date: August 01, 2025
• End date: July 31, 2026
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Algebra
• Principal Investigator: Herivelto Martins Borges Filho
• Grantee: Daniela Alves de Oliveira
• Supervisor: Daniel Panario
• Host Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil
• Institution abroad: Carleton University, Canada
• Associated to the scholarship: 22/14004-7 - Topics in Finite Fields: cyclic codes, Artin-Schreier's hypersufarces and irreducible trinomials, BP.PD

Abstract

In this project, we focus on three topics within the area of Finite Fields. The first objective is the study of properties and enumeration of polynomials over finite fields F_q with specific characteristics. Initially, we explore the enumeration of irreducible monic polynomials of degree n with prescribed coefficients, particularly those that are 1-primitive, 1-normal, or satisfy both properties. We extend existing results to consider the joint conditions of r-primitivity and k-normality for prescribed coefficients. Second, we investigate the stability of polynomial iterates over finite fields, analyzing cubic polynomials, trinomials, and families of higher-degree stable polynomials. Finally, the third topic addresses the finite field analog of the Erd¿s distance problem, analyzing the minimum cardinality of distance sets \Delta(E) for subsets E subset of (F_q)^d. We aim to identify families of sets achieving sharp bounds, relate these problems to combinatorial configurations, and explore their connections to the Fourier transform and exponential sums over finite fields. This research integrates algebraic and combinatorial techniques, providing new insights into polynomial and distance problems in finite fields.