On the rank of Hankel matrices over finite fields

Abstract

Given three nonnegative integers p,q,r and a finite field F, how many Hankel matrices (xi+j)0≤i≤p,0≤j≤q over F have rank ≤r? This question is classical, and the answer (q2r when r≤min⁡{p,q}) has been obtained independently by various authors using different tools ([3, Theorem 1 for m=n], [4, (26)], [5, Theorem 5.1]). In this note, we will study a refinement of this result: We will show that if we fix the first k of the entries x0,x1,…,xk−1 for some k≤r≤min⁡{p,q}, then the number of ways to choose the remaining p+q−k+1 entries xk,xk+1,…,xp+q such that the resulting Hankel matrix (xi+j)0≤i≤p,0≤j≤q has rank ≤r is q2r−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 (and we had to combine some ideas from [4, (26)] and from [5, Theorem 5.1 for r=n] to obtain our proof). The refined result generalizes (and provides an alternative proof of) [1, Corollary 6.4].