Analytically generated sharply o-minimal structures

TL;DR

This paper introduces analytically generated sharply o-minimal structures, providing effective parameterization and preparation theorems that lead to bounds on algebraic points and a proof of Wilkie's conjecture.

Contribution

It develops a new class of sharply o-minimal structures with effective tools for parameterization and preparation, advancing the understanding of definable sets and functions.

Findings

Established a polynomially effective parameterization theorem.

Proved a polynomially effective version of the Yomdin--Gromov lemma.

Derived bounds on algebraic points, confirming Wilkie's conjecture.

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.