Markó Ádám: 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 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. Joint work with Gergely
Kiss, Zoltán Lóránt Nagy, Gábor Somlai.

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