We consider a new theory of positive definiteness – for matrices with entries in a finite field. In the first part of the talk, we will review classical notions of matrix positivity for real and complex matrices as well as background material in algebra and graph theory. We will then examine possible definitions of matrix positivity over finite fields.

In the second part, we will discuss an algebraic version of Schoenberg’s celebrated theorem characterizing the functions $f$ with the property that the matrix $(f(a_{ij}))$ is positive definite for all positive definite matrices $(a_{ij})$. We obtain a complete characterization of such entrywise positivity preservers for matrices of a fixed dimension $d$, over finite fields. For $d \geq 3$, we prove that, surprisingly, the positivity preservers are precisely the positive multiples of the field’s automorphisms. Along the way, we will see 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. (This part is joint with Himanshu Gupta, Prateek Kumar Vishwakarma, and Chi Hoi Yip.)