Séminaire Lotharingien de Combinatoire, 93B.69 (2025), 12 pp.
Dominique Guillot, Himanshu Gupta,
Prateek Kumar Vishwakarma and Chi Hoi Yip
Entrywise Transforms and Positive Definite Matrices over Finite Fields
Abstract.
We consider a natural notion of positive definiteness for matrices over finite fields and prove an algebraic version of Schoenberg's celebrated theorem [Duke Math.~J., 1942] characterizing the functions that preserve positive definiteness when applied entrywise to positive definite matrices. 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 Erdős-Ko-Rado theorem for Paley graphs.
Received: November 15, 2024.
Accepted: February 15, 2025.
Final version: April 1, 2025.
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