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arXiv · 2311.06136

On polynomials of small range sum

Gergely Kiss·Ádám Markó·Zoltán Lóránt Nagy·Gábor Somlai
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Abstract

In order to reprove an old result of R\'edei's on the number of directions determined by a set of cardinality $p$ in $\mathbb{F}_p^2$, Somlai proved that the non-constant polynomials over the field $\mathbb{F}_p$ 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 Lov\'asz and Schrijver using discrete Fourier analysis.

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Gergely Kiss, Ádám Markó, Zoltán Lóránt Nagy, Gábor Somlai. 2023-11-10. On polynomials of small range sum. https://arxiv.org/abs/2311.06136

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