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

TL;DR

This paper characterizes quadratic forms over finite fields and applies these results to explicitly construct maximal and minimal Artin-Schreier curves, advancing understanding of their algebraic and geometric properties.

Contribution

It provides a detailed classification of quadratic forms over finite fields with specific polynomial coefficient restrictions and uses this to explicitly determine maximal and minimal Artin-Schreier curves.

Findings

Explicit characterization of quadratic forms over finite fields.

Construction of maximal Artin-Schreier curves.

Construction of minimal Artin-Schreier curves.

Abstract

This paper explores quadratic forms over finite fields with associated Artin-Schreier curves. Specifically, we investigate quadratic forms of $\mathbb F_{q^n}/\mathbb F_q$ represented by polynomials over $\mathbb F_{q^n}$ 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.