Classification of quadratic forms over finite fields with maximal and minimal Artin-Schreier curves

2024/11/18 by Ruikai Chen, Chen, Ruikai
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2411.11705

openalex publication_date 2024/11/18 · openalex created_date 2024/11/21 · openalex updated_date 2026/07/28

Abstract

This paper explores quadratic forms over finite fields with associated Artin-Schreier curves. Specifically, we investigate quadratic forms of \mathbb F_qⁿ/\mathbb F_q represented by polynomials over \mathbb F_qⁿ with q odd, characterizing them using certain matrices defined by coefficients of the polynomials. In particular, a comprehensive treatment will be given for those polynomials whose coefficients all lie in \mathbb F_q. Afterwards, the results on quadratic forms will be applied to get maximal and minimal Artin-Schreier curves explicitly.

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