$\mathbb{F}_p$ and $Z_p$ Valued Holomorphic Functions over Graphs

TL;DR

This paper explores holomorphic functions over graphs with values in finite fields or cyclic groups, and investigates their relation to complex holomorphic functions on regular trees and p-adic numbers.

Contribution

It introduces a new framework for holomorphic functions over graphs with finite algebraic structures and links them to p-adic analysis and complex holomorphic functions.

Findings

Holomorphic functions over graphs with finite field or cyclic group values are characterized.

A relation between complex holomorphic functions on regular trees and p-adic numbers is established.

New insights into the structure of holomorphic functions in discrete and algebraic settings.

Abstract

The definition of a holomorphic function over a general measurable space $S$ endowed with a Markov process is defined by Zeghib and Barre. In this article we consider holomorphic functions over graphs whose ranges are a given finite field or a cyclic group. Also we consider a relation between $\mathbb{C}$-holomorphic functions over regular trees and the field of $p$-adic numbers.