AI 中文总结
本文研究有限域上多元多项式空间的最大相交族，证明奇数域下所有最大相交族为星集，偶数域下$q\boldsymbol{4}$且$d\boldsymbol{n}$时存在非星型最大例子，还得到非零多项式张成空间及对应线性泛函的相关结果。
AI 中文摘要
设$\boldsymbol{\textit{P}}_{n,d}$为有限域$\boldsymbol{\textit{F}}_q$上$n$个变量、次数不超过$d$的多项式构成的空间。若存在$\boldsymbol{\textit{a}}\boldsymbol{\textit{F}}_q^n$使得$f(\boldsymbol{\textit{a}})=g(\boldsymbol{\textit{a}})$，则称两个多项式$f,g\boldsymbol{\textit{P}}_{n,d}$相交。星集指满足对固定$\boldsymbol{\textit{a}}\boldsymbol{\textit{F}}_q^n$和$b\boldsymbol{\textit{F}}_q$，有$f(\boldsymbol{\textit{a}})=b$的所有$f\boldsymbol{\textit{P}}_{n,d}$构成的集合。本文完全分类$\boldsymbol{\textit{P}}_{n,d}$中的最大相交族。当$n=1$且$d\boldsymbol{2}$时，已有结论表明所有最大相交族均为星集；本文证明，当$q$为奇数时，对所有$n\boldsymbol{2}$且$d\boldsymbol{2}$，该结论同样成立。然而当$q$为偶数时，情况更为微妙，出现了新现象：当$q\boldsymbol{4}$时，仅当$d\boldsymbol{n}$时存在非星型的最大例子。研究过程中，本文还证明了两个具有独立意义的结果：一是确定$\boldsymbol{\textit{P}}_{n,d}$中非零多项式的张成空间；二是刻画所有线性泛函$\boldsymbol{\textit{\textit{P}}}_{n,d}\to\boldsymbol{\textit{F}}_q$，其核与非零多项式集合互不相交。第一个结果是证明本文主要结果的关键；第二个结果是有限域上有界次数多项式的Gleason–Kahane–Żelazko定理。
英文摘要
Let $\mathcal{P}_{n,d}$ be the space of polynomials in $n$ variables over $\mathbb{F}_q$ of degree at most $d$. Two polynomials $f,g\in\mathcal{P}_{n,d}$ intersect if $f(\mathbf a)=g(\mathbf a)$ for some $\mathbf a\in\mathbb{F}_q^n$. A star consists of all polynomials $f\in\mathcal{P}_{n,d}$ satisfying $f(\mathbf a)=b$ for fixed $\mathbf a\in\mathbb{F}_q^n$ and $b\in\mathbb{F}_q$. We completely classify the maximum intersecting families in $\mathcal{P}_{n,d}$. When $n=1$ and $d\geq 2$, it was previously shown that all maximum intersecting families are stars. We prove that the same conclusion holds for all $n\geq 2$ and $d\geq 2$ when $q$ is odd. When $q$ is even, however, the situation is more subtle, and a new phenomenon emerges: for $q\geq 4$, maximum non-star examples exist precisely when $d\leq n$. Along the way, we prove two further results of independent interest. First, we determine the span of nonvanishing polynomials in $\mathcal{P}_{n,d}$. Second, we characterize all linear functionals $Ψ\colon\mathcal{P}_{n,d}\to\mathbb{F}_q$ whose kernels are disjoint from the set of nonvanishing polynomials. The first result plays a crucial role in the proof of our main result; the second is a Gleason--Kahane--Żelazko theorem for polynomials of bounded degree over finite fields.
Comments23 pages
