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Holomorphic functions with Nash real part

Antonio Carbone

TL;DR

This work resolves the question of when a holomorphic function on an open set $D\subset \mathbb{C}^n$ is complex Nash by tying it to the Nash-ness of its real (or imaginary) part. The authors prove that $f=f_1+i f_2$ is complex Nash if and only if $f_1$ (and equivalently $f_2$) is real Nash, using a combination of real–complex algebraic techniques and a Cartan-type construction to reconstruct $f$ from its real part. Key contributions include establishing the equivalence of $f\in \mathcal{N}_{\mathbb{C}}(D)$ with $f_1,f_2\in \mathcal{N}_{\mathbb{R}}(D)$, and providing a self-contained proof that avoids reliance on integration. The results clarify the algebraic structure of Nash holomorphic functions and have implications for definability and algebraicity in several complex variables.

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.

Holomorphic functions with Nash real part

TL;DR

This work resolves the question of when a holomorphic function on an open set is complex Nash by tying it to the Nash-ness of its real (or imaginary) part. The authors prove that is complex Nash if and only if (and equivalently ) is real Nash, using a combination of real–complex algebraic techniques and a Cartan-type construction to reconstruct from its real part. Key contributions include establishing the equivalence of with , and providing a self-contained proof that avoids reliance on integration. The results clarify the algebraic structure of Nash holomorphic functions and have implications for definability and algebraicity in several complex variables.

Abstract

In this paper we show that a holomorphic function, defined on an open subset of , is a complex Nash function if and only if its real part (or equivalently its imaginary part) is a real Nash function.

Paper Structure

This paper contains 5 sections, 3 theorems, 26 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Real and complex Nash functions
  4. Relationship between real and complex Nash functions
  5. Proof of Theorem \ref{['main']}

Key Result

Theorem 1.3

Let $D\subset {\mathbb C}^n$ be an open subset and $f:=f_1+if_2:D\to {\mathbb C}$ a holomorphic function. If $f_1$ is a real Nash function on $D$, then $f$ is a complex Nash function on $D$.

Theorems & Definitions (9)

  • Theorem 1.3
  • Corollary 1.4
  • Definition 2.1
  • Example 2.2
  • Remark 2.3
  • Proposition 2.4
  • proof
  • Remark 2.5
  • proof : Proof of Theorem \ref{['main']}