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arXiv · 2605.02778

$K$-holomorphic functions with definable real part

Antonio Carbone·Enrico Savi
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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.

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Antonio Carbone, Enrico Savi. 2026-05-04. $K$-holomorphic functions with definable real part. https://arxiv.org/abs/2605.02778

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