Wilkie's conjecture for Pfaffian structures

• 成果类型: Article
• 署名作者: Binyamini, Gal; Novikov, Dmitry; Zak, Benny
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2024.199.2.5
• 发表日期: 2024
• 页码: 795-821
• 关键词: rational-points; volume growth; number; field

摘要

1.1. Sharply o-minimal structures. Our general results in this paper are stated in the context of sharply o-minimal structures admitting sharp derivatives. For a complete definition of these notions, see Section 2. Briefly, a sharply o-minimal structure (abbreviated #o-minimal) is a pair (S, S2), where S is an o-minimal expansion of the real field and we identify S with the collection of all definable sets in S; and S2 = {S2 ,D c S} ,DEN is filtration on S depending on the format and degree D. The filtration is increasing with has format and degree D (but note that these are not uniquely defined, as S2 is a filtration rather than a decomposition). One can roughly think of as enco ding the logical complexity,e.g., the length of a formula used to define X, and of D as the sum of the algebraic degrees of all polynomials appearing in such a representation. A system of axioms analogous to the standard axioms of o-minimality controls how the format and degree grows under the standard logical operations, and gives upper bounds on the number of intervals in a definable subset of R in terms of and D.