Holomorphic functions with Nash real part

TL;DR

This paper characterizes holomorphic functions on complex domains as Nash functions precisely when their real or imaginary parts are Nash functions, linking complex analysis with real algebraic geometry.

Contribution

It establishes a necessary and sufficient condition for holomorphic functions to be Nash functions based on their real or imaginary parts.

Findings

Holomorphic functions are Nash iff their real parts are Nash functions.

Provides a characterization connecting complex holomorphic and real Nash functions.

Bridges complex analysis with real algebraic geometry.

Abstract

In this paper we show that a holomorphic function, defined on an open subset $D$ of $\mathbb{C}^n$, is a complex Nash function if and only if its real part (or equivalently its imaginary part) is a real Nash function.