On the rank of Hankel matrices over finite fields

TL;DR

This paper refines the classical count of Hankel matrices over finite fields with rank constraints by fixing initial entries, revealing how these entries influence the total count and connecting to evaluations of Jacobi-Trudi determinants.

Contribution

It provides a refined enumeration of Hankel matrices with rank constraints when initial entries are fixed, extending previous results and offering an alternative proof for related determinant evaluations.

Findings

Number of Hankel matrices with fixed initial entries and rank ≤ r is q^{2r-k}.

The initial entries influence the count in a predictable way, contrary to initial expectations.

The results connect to evaluations of Jacobi-Trudi determinants over finite fields.

Abstract

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.