Totally positive matrices over finite fields: the minimal odd prime for nine shapes, and two corrections

Abstract

Call an element of a finite field positive when it is a nonzero square, and call a matrix over that field totally positive when every one of its minors is positive. Ayyer and Prasad, who introduced this class, ask for the least prime power q admitting an m × n totally positive matrix over 𝔽_(q); over odd q their own data reaches only the shapes 2 × 2 and 3 × 3. We answer the odd-prime half of that question for nine shapes. Two of the three regimes turn out to be classical under other names, and we say so: in characteristic 2 "positive" is "nonzero", so the problem is the existence of a full superregular matrix, equivalently of an [m+n,m,n+1]_q MDS code; and for m=2 an elementary totally positive 2 × n matrix over 𝔽_(q) is exactly a clique of size n+1 in the Paley graph (q ≡ 1 mod 4) or a transitive subtournament of order n+1 in the Paley tournament (q ≡ 3 mod 4), so the 2 × n values reproduce published data. What is left is the odd corner m,n ≥ 3. There we prove that the minimal odd prime is 31 for each of the shapes 3 × 4, 4 × 4 and 3 × 5: the symmetric matrix over 𝔽₃₁ with rows (1,1,1,1), (1,8,5,2), (1,5,9,10), (1,2,10,20) witnesses the first two shapes, an explicit 3 × 5 matrix over 𝔽₃₁ the third, and nothing at all witnesses any of them over 𝔽ₚ for odd p ≤ 29; the non-existence half is an exhaustive search over fewer than 70 000 candidates, made small by a normal form and by the fact that every off-border entry must be a square whose predecessor is also a square. We also record two corrections to the source. Six counts printed there contradict its own Proposition 5.2, which forces |𝔽_(q)⁺|ᵐ⁺ⁿ⁻¹ to divide every such count; the values consistent with that proposition are 236 196, 966 306, 1 492 992 and 141 178 800, 23 640 923 250 000. And the q ≡ 3 (mod 4) half of the proof of its Proposition 5.6 treats the set {a: a, a-1 positive} as totally ordered by "the difference is positive"; at q=19 that set carries the directed 3-cycle 5 → 6 → 17 → 5, and {5,6,17} is precisely the off-border entry set of the 3 × 3 totally positive matrix over 𝔽₁₉ exhibited below. The statement of that proposition is not refuted by anything here. Every numbered result below is machine-checked in Lean 4, except where it is flagged as a paper argument or as a computation made outside the formal development; S[sec:verif] says exactly what the machine checks and what it does not.

Open review

This founding-collection manuscript received AI review before publication. Independent human review is open. Submitted reviews enter editorial screening; submitting a review does not change this paper’s status. Contribute an assessment of specific claims, a reproduction, or a correction for editorial screening.

Archived files

1. Version 1 · current (opens in a new tab)

   Source snapshot 2026-09-07 03:53 UTC

   File fingerprint

   48266e2c3774c19ceacf44bb3dd367bd45f27dc7814f11815402e62f82b5b16e