On polynomials of small range sum

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.