The $\mathbb F_p$-Selberg Integral

TL;DR

This paper establishes a finite field analogue of the Selberg integral, connecting hypergeometric solutions of KZ equations with polynomial solutions modulo a prime p, expanding the understanding of integrals in finite fields.

Contribution

It introduces the $\,\mathbb{F}_p$-Selberg integral formula, bridging hypergeometric solutions and polynomial solutions in finite fields, a novel extension of classical integrals.

Findings

Proves the $\,\mathbb{F}_p$-Selberg integral formula.

Establishes a connection between hypergeometric solutions and polynomial solutions modulo p.

Expands the theory of integrals into finite field contexts.

Abstract

We prove an $\mathbb F_p$-Selberg integral formula, in which the $\mathbb F_p$-Selberg integral is an element of the finite field $\mathbb F_p$ with odd prime number $p$ of elements. The formula is motivated by analogy between multidimensional hypergeometric solutions of the KZ equations and polynomial solutions of the same equations reduced modulo $p$.