arXiv 2003.100402020-03-23math.COcs.ITmath.IT

三次多项式的交分布及相关问题

On the intersection distribution of degree three polynomials and related topics

Gohar Kyureghyan, Shuxing Li, Alexander Pott

备注：: 24 pages, submitted

摘要：

最近Li和Pott（arXiv：2003.06678v1）提出了多项式$f$在有限域$\mathbb{F}_q$上的交集分布，它涉及多项式集合$\{f(x)+cx \mid c \in \mathbb{F}_q\}$的集体行为.相交分布具有潜在的几何解释，其指示$f$的图形与仿射平面$AG(2,q)$中的线之间的相交图案。当$q$是偶数时，长期存在的分类o-多项式的公开问题可以用简单的方式重新表述，即分类所有具有与$x^2$相同的交集分布的多项式。受这种联系的启发，我们继续考虑下一个最简单的情况，并推导出$\mathbb{F}_q$上所有三次多项式的交集分布，其中$q$为奇数和偶数。此外，我们首次将所有具有相同交集分布的单项式分类为$x^3$，得到了这类单项式的一些特征，并提出了一个猜想。此外，还给出了三次多项式的交分布的两个应用。第一个是构造非同构的Steiner三元系，第二个是在仿射平面上产生具有先前未知大小的Kakeya集的无限族。

英文摘要：

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 $\{f(x)+cx \mid c \in \mathbb{F}_q\}$. 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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

相关论文