Journal article
On the rank of Hankel matrices over finite fields
Linear algebra and its applications, v 641, 156
15 May 2022
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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].
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Details
- Title
- On the rank of Hankel matrices over finite fields
- Creators
- Omesh Dhar DwivediDarij Grinberg
- Publication Details
- Linear algebra and its applications, v 641, 156
- Publisher
- Elsevier
- Resource Type
- Journal article
- Language
- English
- Academic Unit
- Mathematics
- Web of Science ID
- WOS:000819926700007
- Scopus ID
- 2-s2.0-85124699803
- Other Identifier
- 991021862366504721
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- Web of Science research areas
- Mathematics
- Mathematics, Applied