Applications of circulant matrices to determinants involving $k$-th power residues

TL;DR

This paper explores the use of circulant matrices and hyperelliptic curves over finite fields to analyze determinants involving Legendre symbols and k-th power residues, revealing new arithmetic properties.

Contribution

It introduces novel methods combining circulant matrices and hyperelliptic curves to study determinants related to power residues over finite fields.

Findings

Derived new arithmetic properties of determinants involving k-th residues.

Established connections between circulant matrices and hyperelliptic curve theory.

Provided insights into the structure of Legendre symbol-based determinants.

Abstract

In this paper, by the tools of circulant matrices and hyperelliptic curves over finite fields, we study some arithmetic properties of certain determinants involving the Legendre symbols and $k$-th residues.