Positive Definite Functions over Finite Fields

• Jarrod Dunn, Mathematics, University of Delaware
• Brian Wilmarth, Mathematics, University of Delaware

• Dominique Guillot, Mathematical Sciences, University of Delaware

A function f: R → R is said to be positive definite if matrices of the form (f(xi − xj))^n_(i,j=1) are positive definite for all choices of distinct x_1, x_2, . . . , x_n ∈ R and all n ≥ 2. Such functions naturally appear in many areas such as mathematical analysis, machine learning, probability and statistics, and physics. By a famous theorem of Bochner, positive definite functions can be identified with positive measures on the real line, via the Fourier transform.

Recently, researchers have considered notions of positive definiteness for matrices over finite fields. One such important example is Z_p, the integers modulo a prime number p. Understanding positive definite functions in that setting is a natural open problem with possible important implications in areas such as coding theory and cryptography, where finite fields are widely used.

Our investigation using symbolic calculations suggests that, up to rescaling, only one function is positive definite over Z_p: the function f(x) = 1+(p−1)x^(p−1) which maps 0 to 1 and all other field elements to 0. This was rigorously tested for several small values of p by enumerating and testing all possible functions. This outcome contrasts sharply with the real-number case, where many functions are positive definite. The results also highlight the rigidity of finite fields and the strong algebraic constraints they impose on matrix transformations. For non-prime finite fields, we observed non-trivial positive definite functions. Their general structure appears to be more intricate and is currently not well understood.