Notes on $2D$ $\mathbb F_p$-Selberg integrals

TL;DR

This paper establishes a two-dimensional finite field analogue of the Selberg integral, linking hypergeometric solutions of KZ equations to polynomial solutions modulo a prime, expanding the understanding of integrals in finite fields.

Contribution

It introduces a novel two-dimensional $ ext{F}_p$-Selberg integral formula, bridging hypergeometric solutions of KZ equations and polynomial solutions in finite fields.

Findings

Proves a new two-dimensional $ ext{F}_p$-Selberg integral formula.

Connects hypergeometric solutions of KZ equations with polynomial solutions modulo $p$.

Provides a foundation for further exploration of finite field integrals.

Abstract

We prove a two-dimensional $\mathbb F_p$-Selberg integral formula, in which the two-dimensional $\mathbb F_p$-Selberg integral $\bar S(a,b,c;l_1,l_2)$ depends on positive integer parameters $a,b,c$, $l_1,l_2$ and is an element of the finite field $\mathbb F_p$ with odd prime number $p$ of elements. The formula is motivated by the analogy between multidimensional hypergeometric solutions of the KZ equations and polynomial solutions of the same equations reduced modulo $p$.