Analytically generated sharply o-minimal structures
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866915921460723712 |
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| author | Carmon, Oded |
| author_facet | Carmon, Oded |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_06144 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Analytically generated sharply o-minimal structures Carmon, Oded Logic Algebraic Geometry Number Theory 14P10, 03C64, 11G99, 11U09 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. |
| title | Analytically generated sharply o-minimal structures |
| topic | Logic Algebraic Geometry Number Theory 14P10, 03C64, 11G99, 11U09 |
| url | https://arxiv.org/abs/2604.06144 |