有限域上Hankel矩阵的秩研究
On the rank of Hankel matrices over finite fields
AI总结：
本研究细化了有限域上秩不超过r的Hankel矩阵计数的经典结果，证明固定前k个元素时满足秩条件的剩余元素选择数为q^(2r−k)，并推广了有限域上Jacobi-Trudi行列式求值的相关结论。
摘要：
给定三个非负整数$p,q,r$和一个有限域$F$，域$F$上秩不超过$r$的Hankel矩阵$\bigl( x_{i+j}\bigr) _{0\leq i\leq p,\\ 0\leq j\leq q}$共有多少个？这是一个经典问题，其答案（当$r\leq\min\left\{ p,q\right\} $时为$q^{2r}$）已由多位学者（Daykin、Elkies、Garcia Armas、Ghorpade和Ram）使用不同工具独立得到。在本注记中，我们研究该结果的一个细化情形：我们证明，若对某个满足$k\leq r\leq\min\left\{ p,q\right\} $的$k$，固定前$k$个元素$x_{0},x_{1},\ldots,x_{k-1}$，则选择剩余$p+q-k+1$个元素$x_{k},x_{k+1},\ldots,x_{p+q}$，使得所得Hankel矩阵$\bigl( x_{i+j}\bigr) _{0\leq i\leq p,\\ 0\leq j\leq q}$的秩不超过$r$的方式数为$q^{2r-k}$。这恰好是人们假设前$k$个元素对秩无影响时会得到的答案，但实际情况显然并非如此简单。该细化结果推广了（并提供了另一种证明方式）Anzis、Chen、Gao、Kim、Li和Patrias关于有限域上Jacobi-Trudi行列式求值的一项结果。
英文摘要：
Given three nonnegative integers $p,q,r$ and a finite field $F$, how many Hankel matrices $\left( x_{i+j}\right) _{0\leq i\leq p,\ 0\leq j\leq q}$ over $F$ have rank $\leq r$ ? This question is classical, and the answer ($q^{2r}$ when $r\leq\min\left\{ p,q\right\} $) has been obtained independently by various authors using different tools (Daykin, Elkies, Garcia Armas, Ghorpade and Ram). In this note, we study a refinement of this result: We show that if we fix the first $k$ of the entries $x_{0},x_{1},\ldots,x_{k-1}$ for some $k\leq r\leq\min\left\{ p,q\right\} $, then the number of ways to choose the remaining $p+q-k+1$ entries $x_{k},x_{k+1},\ldots,x_{p+q}$ such that the resulting Hankel matrix $\left( x_{i+j}\right) _{0\leq i\leq p,\ 0\leq j\leq q}$ has rank $\leq r$ is $q^{2r-k}$. This is exactly the answer that one would expect if the first $k$ entries had no effect on the rank, but of course the situation is not this simple. The refined result generalizes (and provides an alternative proof of) a result by Anzis, Chen, Gao, Kim, Li and Patrias on evaluations of Jacobi-Trudi determinants over finite fields.
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