Minimal field size for totally positive matrices

Determine, for fixed positive integers m and n, the minimal prime power q for which an m by n totally positive matrix exists over F_q, and characterize how this minimal q grows with m and n.

Background

The paper derives upper bounds on the matrix dimensions permitted by a finite field for the existence of totally positive matrices. The authors note that their argument uses only certain minors of order two and therefore is likely far from optimal.

They explicitly pose the problem of finding the exact threshold field size and determining its asymptotic growth as the dimensions increase.

References

This motivates the following question. For fixed positive integers $m$ and $n$, find the minimal prime power $q$ such that there exists an $m \times n$ totally positive matrix over $\mathbb{F}_q$. In particular how does this minimal value of $q$ grow with $m$ and $n$?

— Positive definite, positive semidefinite and totally positive matrices over finite fields  (2608.17702 - Ayyer et al., 18 Aug 2026) in Question following Proposition 2.??, subsection “Bounds relating the order and q” of Section “Totally positive matrices”

By reasoning similar to that in \cref{rem:when psd 5 polyn} and the formulas for the number of totally positive matrices for small orders given in \cref{sec:totpos enum}, we make the following conjecture. For a prime $p$ and positive integers $m, n$, there exists a positive integer $t$ and rational polynomials $g_0(x),\dots,g_{t-1}(x)\in \mathbb{Q}[x]$ of degree $m n$, such that the number of totally positive matrices in $M_{m,n}(\mathbb{F}_{pk})$ is $g_i(pk)$ when $k\equiv i\pmod{t}$.

— Positive definite, positive semidefinite and totally positive matrices over finite fields  (2608.17702 - Ayyer et al., 18 Aug 2026) in Conjecture following Theorem 2.??, Section “Structural formula” of Section “Totally positive matrices”