Square root Bound on the Least Power Non-residue using a Sylvester-Vandermonde Determinant

TL;DR

This paper presents an elementary proof establishing a bound on the least k-th power non-residue in an arithmetic progression over a prime field, utilizing a Sylvester-Vandermonde determinant and Stepanov's method.

Contribution

It introduces a novel elementary proof technique combining determinant identities and the Stepanov method to bound non-residues in prime fields.

Findings

Bound on least k-th power non-residue in arithmetic progressions.

Utilizes Sylvester-Vandermonde determinant in number theory.

Provides an elementary alternative to existing proofs.

Abstract

We give a new elementary proof of the fact that the value of the least $k^{th}$ power non-residue in an arithmetic progression $\{bn+c\}_{n=0,1...}$, over a prime field $\F_p$, is bounded by $7/\sqrt{5} \cdot b \cdot \sqrt{p/k} + 4b + c$. Our proof is inspired by the so called \emph{Stepanov method}, which involves bounding the size of the solution set of a system of equations by constructing a non-zero low degree auxiliary polynomial that vanishes with high multiplicity on the solution set. The proof uses basic algebra and number theory along with a determinant identity that generalizes both the Sylvester and the Vandermonde determinant.