On the Intersection Distribution of Degree Three Polynomials and Related Topics

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.

On two conjectures about the intersection distribution

“…Recently, G. Kyureghyan, et al [6] proceeded to consider the next simplest case and derive the intersection distribution for all degree three polynomials over F q with q both odd and even. Moreover, they initiated to classify all monomials having the same intersection distribution as x 3 , where some characterizations of such monomials were obtained and some conjectures were proposed.…”

“…In this paper, we completely solve Conjecture 1. According to Theorem 3.8 of [6] (see also Lemma 2), the key point of proving Conjecture 1 is to establish that some polynomial g d (x) = x d −1…”

“…In this paper, we completely solve two conjectures in [6]. Namely, we prove two classes of power functions having intersection distribution: v 0 (f ) = q(q−1)…”

“…Later, G. Kyureghyan, et al [6] proceeded to consider the next simplest case and derive the intersection distribution for all degree three polynomials over F q with q both odd and even. They also proposed several conjectures in [6].…”

“…Later, G. Kyureghyan, et al [6] proceeded to consider the next simplest case and derive the intersection distribution for all degree three polynomials over F q with q both odd and even. They also proposed several conjectures in [6].In this paper, we completely solve two conjectures in [6]. Namely, we prove two classes of power functions having intersection distribution: v 0 (f ) = q(q−1)We mainly make use of the multivariate method and QM-equivalence on 2-to-1 mappings.…”