New bound on small range sum polynomials of degree

TL;DR

This paper improves bounds on small-range sum polynomials of degree (p-1)/2 over primes, extending previous results to smaller primes using elementary Legendre symbol sum estimates.

Contribution

It introduces a new elementary method for estimating Legendre symbol sums, reducing the prime bound from large to as small as 23.

Findings

Extended the bound on small-range sum polynomials to primes as small as 23

Developed a new elementary approach for Legendre symbol sum estimation

Achieved tighter bounds compared to previous large prime requirements

Abstract

The polynomials of degree $\frac{p-1}{2}$ of range sum $p$ was determined in {\tt arXiv:2311.06136 [math.NT]} for large enough primes. We extend this result by reducing the lower bound for the primes to $23$ by introducing a new and elementary way of estimating sums of Legendre symbols.