On the rank of Hankel matrices over finite fields

Given three nonnegative integers p, q, r and a finite field F, how many Hankel matrices (x(i+j))(0 <= i <= p), (0 <= j <= q) over F have rank <= r? This question is classical, and the answer (q(2r) 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 x(0), x(1), ..., x(k-1) for some k <= r <= min {p, q}, then the number of ways to choose the remaining p + q - k + 1 entries x(k), x(k+1), ..., x(p+q) such that the resulting Hankel matrix (x(i+j))(0 <= i <= p), (0 <= j <= q) has rank <= r is q(2r-k). This is exactly the answer thiat 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]. (C) 2022 Elsevier Inc. All rights reserved.