On polynomials of small range sum

In order to reprove an old result of Rédei on the number of directions determined by a set of cardinality p in Fp², Somlai proved thatnon-constant polynomials over the field Fp whose range sum is equal to p are of degree at least (p-1)/(2). Here the values arerepresented by integers in [0, p − 1]. In this paper, we characterise all of these polynomials having degree exactly (p-1)/(2), if p islarge 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.