Abstract
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.
Highlights
Throughout this paper, let Fq = Fpm be a finite field with characteristic p and f a polynomial over Fq
The intersection distribution of f originates from an elementary problem concerning the interaction between the graph {(x, f (x)) | x ∈ Fq} of f and the lines in the classical affine plane AG(2, q)
For 0 i q, we ask about the number of affine lines intersecting the graph of f in exactly i points
Summary
The aforementioned geometric interpretation indicates that for the point set in the classical projective plane P G(2, q) arising from a polynomial f , detailed information follows from the intersection distribution f (see for instance [22, Proposition 3.2]). We remark that in general, computing the intersection and multiplicity distributions is a nontrivial problem. We remark that f having degree two forces vi(f ) = 0 for each i > 2, so that the intersection distribution (1.1) follows from Proposition 1.2. Together with Proposition 1.2, the intersection distribution of each degree three polynomial f can be determined by exactly one of vi(f ), 0 i 3.
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