$K$-holomorphic functions with definable real part
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866910189362348032 |
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| author | Carbone, Antonio Savi, Enrico |
| author_facet | Carbone, Antonio Savi, Enrico |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_02778 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | $K$-holomorphic functions with definable real part Carbone, Antonio Savi, Enrico Algebraic Geometry Logic Primary: 03C64, 14P10, Secondary: 14P20, 26E05, 32B20 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. |
| title | $K$-holomorphic functions with definable real part |
| topic | Algebraic Geometry Logic Primary: 03C64, 14P10, Secondary: 14P20, 26E05, 32B20 |
| url | https://arxiv.org/abs/2605.02778 |