On polynomials of small range sum

Abstract

In order to reprove an old result of Rédei'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 Lovász and Schrijver using discrete Fourier analysis.

arXiv commentThe 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