On the Intersection Distribution of Degree Three Polynomials and Related Topics

The intersection distribution of a polynomial $f$ over finite field $\mathbb{F}_q$ was recently proposed by Li and Pott [\emph{Finite Fields and Their Applications, 66 (2020)}], which concerns the collective behaviour of a collection of polynomials $\{f(x)+cx \mid c \in\mathbb{F}_q\}$. The intersection distribution has an underlying geometric interpretation, which indicates the intersection pattern between the graph of $f$ and the lines in the affine plane $AG(2,q)$. When $q$ is even, the long-standing open problem of classifying o-polynomials can be rephrased in a simple way, namely, classifying all polynomials which have the same intersection distribution as $x^2$. Inspired by this connection, we proceed to consider the next simplest case and derive the intersection distribution for all degree three polynomials over $\mathbb{F}_q$ with $q$ both odd and even. Moreover, we initiate to classify all monomials having the same intersection distribution as $x^3$, where some characterizations of such monomials are obtained and a conjecture is proposed. In addition, two applications of the intersection distributions of degree three polynomials are presented. The first one is the construction of nonisomorphic Steiner triple systems and the second one produces infinite families of Kakeya sets in affine planes with previously unknown sizes.

Point Distribution and Perfect Directions in 𝔽p2\mathbb{F}_p^2

Let p ≥ 3 be a prime, S \subseteq \mathbb{F}_p^2 a nonempty set, and w:\mathbb{F}_p^2 \to R a function with supp w = S. Applying an uncertainty inequality due to András Bíró and the present author, we show that there are at most {1 \over 2}\left| S \right| directions in \mathbb{F}_p^2 such that for every line l in any of these directions, one has \sum\limits_{z \in l} {w\left( z \right) = {1 \over p}\sum\limits_{z \in \mathbb{F}_p^2} {w\left( z \right),} } except if S itself is a line and w is constant on S (in which case all, but one direction have the property in question). The bound {1 \over 2}\left| S \right| is sharp.As an application, we give a new proof of a result of Rédei-Megyesi about the number of directions determined by a set in a finite affine plane.

Frameworks for Two-Dimensional Keller Maps

The classical Jacobian Conjecture asserts that every locally invertible polynomial self-map of the complex affine space is globally invertible. A Keller map is a (hypothetical) counterexample to the Jacobian Conjecture. In dimension two every such map, if exists, leads to a map between the Picard groups of suitable compactifications of the affine plane, that satisfy a complicated set of conditions. This is essentially a combinatorial problem. Several solutions to it ("frameworks") are described in detail. Each framework corresponds to a large system of equations, whose solution would lead to a Keller map.