arxiv: 2605.02778 · v1 · submitted 2026-05-04 · 🧮 math.AG · math.LO

Recognition: unknown

K-holomorphic functions with definable real part

Antonio Carbone, Enrico Savi

Pith reviewed 2026-05-08 17:26 UTC · model grok-4.3

classification 🧮 math.AG math.LO

keywords K-holomorphic functionsdefinable real parto-minimal structuressemialgebraic setsstrong R-analyticityreal closed fieldscomplex analysis

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The pith

K-holomorphic functions on definable open sets have definable real parts precisely when those real parts are strongly R-analytic, at least in the semialgebraic case.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies K-holomorphic functions f = f1 + i f2 defined on open definable subsets U of K^n, where K is the algebraic closure of a real closed field R, in a fixed o-minimal structure. It establishes the precise relationship between the definability of f and the definability together with strong R-analyticity of the real part f1. A complete characterisation of this relationship is obtained when the structure is semialgebraic, with the results shown to be optimal in general and dependent on the particular o-minimal structure. A sympathetic reader would care because the characterisation reduces questions about complex definable functions to properties of their real parts alone within tame geometric settings.

Core claim

In the semialgebraic case, a K-holomorphic function f on an open definable set U subset K^n is definable if and only if its real part f1 is both definable and strongly R-analytic.

What carries the argument

The equivalence linking definability of the full K-holomorphic function to definability and strong R-analyticity of its real part, within semialgebraic o-minimal structures on real closed fields.

If this is right

Definable K-holomorphic functions on semialgebraic sets necessarily have strongly R-analytic real parts.
Verification of both definability and K-holomorphicity reduces to checking the real part alone in the semialgebraic setting.
The imaginary part need not be examined separately once the real part satisfies the strong analyticity condition.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Similar characterisations may hold in other o-minimal structures if additional tameness conditions are imposed.
The result suggests a route to classifying definable complex analytic objects by examining only real components in real algebraic geometry.
Counterexamples outside the semialgebraic case could be constructed to test the sharpness of the stated optimality.

Load-bearing premise

The o-minimal structure under consideration is semialgebraic, since the completeness and precision of the characterisation vary across different o-minimal structures.

What would settle it

A concrete K-holomorphic function on a semialgebraic open set U whose real part is definable yet fails to be strongly R-analytic, or whose full function is definable while the real part is not strongly R-analytic.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a clean complete characterization in the semialgebraic case for when a K-holomorphic function has a definable real part, tying it directly to definability plus strong R-analyticity of that part.

read the letter

The main takeaway is that in the semialgebraic setting they fully pin down the link: a K-holomorphic f on a definable open set has definable real part precisely when the real part itself is definable and strongly R-analytic. That characterization looks new and is stated as such. They also give general results showing that definability of f forces the real part to satisfy those conditions, with the converse holding in ways that depend on the o-minimal structure. The authors flag upfront that the results are best possible in general and that precision varies with the structure, which keeps the claims honest and avoids overreach. The work extends standard o-minimal techniques to this holomorphic setting without visible circularity or hidden fitting. The derivations rest on ordinary properties of holomorphic functions over real closed fields and definable sets, so the logic tracks. The soft spot is the narrow scope. Everything stays inside definable sets in o-minimal expansions, so the results do not speak to holomorphic functions outside that framework or to non-definable cases. That is not a flaw in the execution, just a limit on how far the statements travel. This is for people already working in o-minimal geometry or real algebraic geometry with complex-valued functions. A reader in that niche gets a concrete, usable characterization in the semialgebraic case and clear optimality statements. It deserves peer review because the semialgebraic part is specific enough to check directly and the general claims are framed with appropriate caution.

Referee Report

0 major / 2 minor

Summary. The paper examines the relationship between the definability of a K-holomorphic function f = f1 + i f2 : U → K, where U is an open definable set in an o-minimal structure over a real closed field R with K = R(i), and the definability together with strong R-analyticity of its real part f1. It establishes a complete characterization of this relationship in the semialgebraic case and asserts that the results are best possible in general, with their precision depending on the specific o-minimal structure considered.

Significance. If the derivations hold, the work supplies a precise and optimal bridge between K-holomorphicity and real definability in o-minimal geometry, particularly through the complete semialgebraic characterization. This strengthens the toolkit for studying definable complex functions and could support further results on analytic continuation or definable sets in real closed fields.

minor comments (2)

The abstract and introduction would benefit from a brief explicit statement of the o-minimal structure axioms used in the semialgebraic characterization (e.g., which closure properties are invoked in the proof of the main theorem).
Notation for 'strong R-analyticity' is introduced without a dedicated preliminary subsection; a short definition or reference to its distinction from ordinary real-analyticity would improve readability for readers outside o-minimal geometry.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and for recommending minor revision. The referee's summary accurately describes the paper's focus on the relationship between definability of K-holomorphic functions and the definability plus strong R-analyticity of their real parts, including the complete characterization in the semialgebraic case. No specific major comments were listed in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper derives a complete semialgebraic characterization relating definability of a K-holomorphic function f to definability and strong R-analyticity of its real part f1, using standard properties of o-minimal structures on real closed fields. No step reduces by construction to a fitted input, self-definition, or load-bearing self-citation chain; the claims rest on external mathematical facts about definable sets and holomorphic functions rather than renaming or smuggling ansatzes. The abstract explicitly notes that precision depends on the o-minimal structure and results are best possible in general, confirming the analysis is independent and falsifiable against the structure's axioms.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard background axioms of o-minimal structures, real closed fields, and holomorphic functions over their algebraic closures. No free parameters are introduced or fitted, and no new entities are postulated.

axioms (2)

standard math Standard properties of o-minimal structures (cell decomposition, monotonicity, etc.) and real closed fields
Invoked throughout as the ambient setting for definable sets and functions.
standard math Basic facts about K-holomorphic functions and strong R-analyticity
Used to relate the complex and real parts.

pith-pipeline@v0.9.0 · 5404 in / 1401 out tokens · 49125 ms · 2026-05-08T17:26:14.143168+00:00 · methodology

discussion (0)

Reference graph

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