论文标题
关于三个多项式和相关主题的交点分布
On the intersection distribution of degree three polynomials and related topics
论文作者
论文摘要
The intersection distribution of a polynomial $f$ over finite field $\mathbb{F}_q$ was recently proposed in Li and Pott (arXiv:2003.06678v1), which concerns the collective behaviour of a collection of polynomials $\{f(x)+cx \mid c \in \mathbb{F}_q\}$.交叉分布具有基本的几何解释，该解释表明$ f $的图与仿射平面$ ag（2，q）$中的线之间的相交模式。当$ q $甚至是$ q $时，可以简单的方式对o polynomials进行分类的长期开放问题，即，对所有具有与$ x^2 $相同的交叉分布的多项式分类。受此连接的启发，我们继续考虑下一个最简单的情况，并在$ \ mathbb {f} _q $上带有$ q $的所有程度上的三个多项式的交叉分布，均奇数甚至奇数。此外，我们开始对所有具有与$ x^3 $相同的交叉分布的单一元素进行分类，其中获得了此类单元的某些特征并提出了猜想。此外，提出了三个多项式的交点分布的两个应用。第一个是构造非构形式施泰纳三重系统，第二个是在以前未知尺寸的仿射平面中生产无限的Kakeya家族。
The intersection distribution of a polynomial $f$ over finite field $\mathbb{F}_q$ was recently proposed in Li and Pott (arXiv:2003.06678v1), 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.
