$K$-holomorphic functions with definable real part

TL;DR

This paper explores the connection between definability of K-holomorphic functions and their real parts within o-minimal structures, providing optimal results and a full characterization in the semialgebraic case.

Contribution

It establishes the precise relationship between definability of K-holomorphic functions and their real parts, with a complete characterization in the semialgebraic setting.

Findings

Definability of a K-holomorphic function relates to the definability and R-analyticity of its real part.

Results are optimal and depend on the o-minimal structure considered.

Complete characterization achieved in the semialgebraic case.

Abstract

Let $R$ be a real closed field and $K:=R(i)$ its algebraic closure. Let $U\subset K^n$ be an open and definable set in a fixed o-minimal structure. In this note, we study the relationship between definability of a $K$-holomorphic function $f=f_1+if_2:U\to K$ and the definability and (strong) $R$-analyticity of its real part $f_1:U\to R$. Our results turn out to be the best possible {in general}, and their precision depends on the considered o-minimal structure. We obtain a complete characterisation in the semialgebraic case.