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 in the significantly more challenging dimension 2 case over a finite field with q elements, unless $q\equiv 1\left(\mathrm{mod}4\right)$ 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-Kovács, and via the structure of cliques in Paley graphs.