Degree gap of polynomials of small range sum

TL;DR

This paper investigates the existence of polynomials with specific degrees and range sums over finite fields, establishing non-existence results for certain degrees when the prime is large enough.

Contribution

It proves that no polynomial of degree (p+1)/2 with range sum p exists for sufficiently large primes, extending previous work on polynomials of degree (p-1)/2.

Findings

Polynomials of degree (p-1)/2 with range sum p are characterized.

No polynomials of degree (p+1)/2 with range sum p exist for large primes.

The results depend on properties of finite fields and prime size.

Abstract

Polynomials of degree $\frac{p-1}{2}$ of range sum $p$ were determined by the second authors relying on a joint work of the authors by Kiss and Nagy. We prove that for large enough primes there is no polynomial of degree $\frac{p+1}{2}$ of range sum $p$.