On polynomials of small range sum

Gergely Kiss; ÃdÃ¡m MarkÃ³; ZoltÃ¡n LÃ³rÃ¡nt Nagy; GÃ¡bor Somlai

number theory

On polynomials of small range sum

arXiv:2311.06136

summary

The paper classifies all non‑constant polynomials over the finite field $\mathbb{F}_p$ whose integer range sums equal $p$ and have degree exactly $(p-1)/2$ for sufficiently large primes, and uses this to re‑derive a known result about sets with few determined directions via discrete Fourier analysis.

Abstract

In order to reprove an old result of RÃ©dei's on the number of directions determined by a set of cardinality $p$ in $F_{p}^{2}$ , Somlai proved that the non-constant polynomials over the field $F_{p}$ whose range sums are equal to $p$ are of degree at least $\frac{p - 1}{2}$ . Here the summand in the range sum are considered as integers from the interval $[0, p - 1]$ . In this paper we characterise all of these polynomials having degree exactly $\frac{p - 1}{2}$ , if $p$ is large enough. As a consequence, for the same set of primes we re-establish the characterisation of sets with few determined directions due to LovÃ¡sz and Schrijver using discrete Fourier analysis.

The manuscript was accepted by ARS MATHEMATICA CONTEMPORANEA. During the review process, several inaccuracies were identified in the statements and proofs, for which we are very grateful, but the main result, of course, remains unchanged

Topics & keywords

#finite fields #polynomial degree #range sum #direction problem #discrete fourier analysisF_prange sumpolynomial degree boundRédei theoremLovász‑SchrijverFourier analysis