Positivity preservers over finite fields

Abstract

We resolve an algebraic version of Schoenberg's celebrated theorem [Duke Math.J., 1942] characterizing entrywise matrix transforms that preserve positive definiteness. Compared to the classical real and complex settings, we consider matrices with entries in a finite field and obtain a complete characterization of such preservers for matrices of a fixed dimension. When the dimension of the matrices is at least $3$, we prove that, surprisingly, the positivity preservers are precisely the positive multiples of the field's automorphisms. We also obtain characterizations of preservers for matrices of dimension $2$ over a finite field with $q$ elements, unless $q \equiv 1 \pmod 4$ and $q$ is not a square. Our proofs build on several novel connections between positivity preservers and field automorphisms via the works of Weil, Carlitz, and Muzychuk-Kov\'acs, and via the structure of cliques in Paley graphs.