Positivity preservers over finite fields

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. Our work makes crucial use of the well-known character-sum bound due to Weil, and of a result of Carlitz Proc. Amer. Math. Soc. , 1960 that provides a characterization of the automorphisms of Paley graphs.