  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 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.
