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arxiv: 2604.06144 · v1 · submitted 2026-04-07 · 🧮 math.LO · math.AG· math.NT

Analytically generated sharply o-minimal structures

Oded Carmon This is my paper

Pith reviewed 2026-05-10 18:18 UTC · model grok-4.3

classification 🧮 math.LO math.AGmath.NT
keywords o-minimal structuresWilkie's conjectureYomdin-Gromov lemmaparameterization theorempreparation theoremcomplex cellssharply o-minimaldefinable sets
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The pith

Analytically generated sharply o-minimal structures admit polynomially effective parameterization theorems that imply Wilkie's conjecture.

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

The paper defines a class of sharply o-minimal structures called analytically generated ones, in which definable sets and their complexity are fixed by a collection of definable complex cells. It establishes a polynomially effective parameterization theorem for real definable sets using real complex cells. This yields a polynomially effective form of the Yomdin-Gromov lemma on C^r-smooth parameterizations of definable sets. The lemma in turn produces polylogarithmic bounds on the number of algebraic points of bounded height and degree that lie in the transcendental part of any definable set, which is exactly Wilkie's conjecture. The same approach also delivers a polynomially effective preparation theorem for definable functions, analogous to known results in the subanalytic case.

Core claim

In analytically generated sharply o-minimal structures, definable sets and their complexity filtration are determined by the collection of definable complex cells. This property permits a polynomially effective parameterization theorem for real sets using real complex cells. The theorem yields a polynomially effective Yomdin-Gromov lemma on C^r-smooth parameterizations of definable sets, which implies Wilkie's conjecture on polylogarithmic bounds for algebraic points of bounded height and degree in the transcendental part of a definable set. It also produces a polynomially effective preparation theorem for definable functions.

What carries the argument

Analytically generated structures, whose definable sets and complexity filtration are determined by the collection of definable complex cells.

If this is right

  • A polynomially effective Yomdin-Gromov lemma holds for C^r-smooth parameterizations of definable sets.
  • Wilkie's conjecture is satisfied: the number of algebraic points of bounded height and degree in the transcendental part of any definable set is bounded by a polylogarithmic function.
  • A polynomially effective preparation theorem holds for definable functions, comparable to the subanalytic preparation theorems.
  • The parameterization theorem applies directly to real sets definable in the structure using real complex cells.

Where Pith is reading between the lines

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

  • If other o-minimal structures can be shown to be analytically generated, the same effective bounds would apply to them without further work.
  • The complex-cell filtration may allow similar effective results in related tame geometries where complex cells are already well understood.
  • The polynomial effectiveness could be used to obtain explicit constants in point-counting applications that previously had only asymptotic statements.

Load-bearing premise

The structure must be analytically generated, so that all definable sets and their complexity are completely fixed by definable complex cells.

What would settle it

An explicit example of an analytically generated structure containing a definable set whose transcendental part has more than polylogarithmically many algebraic points of bounded height and degree would refute the central implication.

Figures

Figures reproduced from arXiv: 2604.06144 by Oded Carmon.

Figure 1
Figure 1. Figure 1: Clustering sections of the projection π : Z → C1..ℓ. Above each point z ∈ C1..ℓ, the complement of the sections π −1 (z) is covered by discs and annuli whose extensions do not intersect π −1 (z). Annuli centered around one of the sections (e.g. the origin) group the remaining sections into clusters. The region between two such annuli is covered by discs and by similar configurations of discs and annuli cen… view at source ↗
Figure 2
Figure 2. Figure 2: Condition on a cell with non-zero center. Since the 1/3- extension of the disc D(r) + θj,k does not meet the origin, we have that |xk − θj,k| < r and 2r < |xk| for all xk ∈ A(·, r) + θj,k . Hence |xk − θj,k| < 1 2 |xk|. In general, for a prepared map xk = z qk k + θj,k and a δ-extension, we have |xk − θj,k| < δ qk 1−δ qk |xk|. In addition, if θj,k is not identically 0 over φj (R+Cj ), then it is nowhere va… view at source ↗
read the original abstract

We describe a class of sharply o-minimal structures, called analytically generated structures, whose definable sets and their complexity filtration are determined by the collection of definable complex cells. We prove a polynomially effective parameterization theorem using real complex cells for real sets definable in such structures. Following Binyamini--Novikov, this allows us to establish a polynomially effective version of the Yomdin--Gromov lemma on C^r-smooth parameterizations of definable sets, which implies Wilkie's conjecture on polylogarithmic bounds for the amount of algebraic points of bounded height and degree in the transcendental part of a definable set. In addition, we obtain a polynomially effective preparation theorem for definable functions, similar to the subanalytic preparation theorems of Parusinski and of Lion--Rolin.

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.

Referee Report

0 major / 2 minor

Summary. The paper introduces analytically generated sharply o-minimal structures, defined so that definable sets and their complexity filtration are determined by the collection of definable complex cells. It proves a polynomially effective parameterization theorem for real definable sets using real complex cells. Following Binyamini--Novikov, this yields a polynomially effective Yomdin--Gromov lemma on C^r-smooth parameterizations, implying Wilkie's conjecture on polylogarithmic bounds for algebraic points of bounded height and degree in the transcendental part of definable sets. It also establishes a polynomially effective preparation theorem for definable functions, analogous to subanalytic results of Parusiński and Lion--Rolin.

Significance. If the results hold, the work is significant for providing effective (polynomial) control in a new class of o-minimal structures without exponential losses, directly enabling an effective Yomdin--Gromov lemma and a resolution of Wilkie's conjecture. The definition via complex cells ensures the filtration supports these bounds, building cleanly on prior work while delivering falsifiable effective statements with potential applications in Diophantine geometry.

minor comments (2)
  1. [3] Section 3: the parameterization theorem would benefit from an explicit statement of how the polynomial degree depends on the complexity of the input complex cell decomposition.
  2. [4] Section 4: the preparation theorem statement could include a brief remark confirming that no hidden non-polynomial factors arise from the transfer between complex and real cells.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful and positive report, which accurately summarizes the main results on analytically generated sharply o-minimal structures, the polynomially effective parameterization theorem, the Yomdin-Gromov lemma, and the preparation theorem. The recommendation for minor revision is noted.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained from definition

full rationale

The paper introduces analytically generated structures by defining their definable sets and complexity filtration explicitly in terms of definable complex cells. It then proves a polynomially effective parameterization theorem for real sets in these structures, followed by an effective Yomdin-Gromov lemma via external appeal to Binyamini-Novikov. No step reduces a claimed result to a fitted parameter, self-citation chain, or definitional tautology; the central implications for Wilkie's conjecture follow from the new definition plus standard o-minimality without internal reduction to inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the new definition of analytically generated structures together with standard background axioms of o-minimality and properties of complex cells from algebraic geometry. No numerical free parameters appear; the results are theorem statements rather than fitted quantities.

axioms (2)
  • domain assumption Standard axioms of o-minimality
    The paper works inside the framework of o-minimal structures from model theory.
  • domain assumption Definability and filtration properties of complex cells
    Relies on prior results about definable complex cells in complex geometry.

pith-pipeline@v0.9.0 · 5435 in / 1503 out tokens · 73323 ms · 2026-05-10T18:18:17.410148+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

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