Counting Algebraic Points in Expansions of O-Minimal Structures by a Dense Set
Counting Algebraic Points in Expansions of O-Minimal Structures by a Dense Set
复制标题
计算稠密集 O-最小结构展开中的代数点
DOI：
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发表时间：
2017
影响因子：
0.7
通讯作者：
Pantelis E. Eleftheriou
中科院分区：
文献类型：
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作者：
Pantelis E. Eleftheriou
The Pila–Wilkie theorem states that if a set $Xsubseteq mathbb{R}^n$ is definable in an o-minimal structure $mathcal{R}$ and contains ‘many’ rational points, then it contains an infinite semialgebraic set. In this paper, we extend this theorem to an expansion $widetilde{mathcal{R}}=langle {mathcal{R}}, P
angle$ of ${mathcal{R}}$ by a dense set P, which is either an elementary substructure of ${mathcal{R}}$, or it is $mathrm{dcl}$-independent, as follows. If X is definable in $widetilde{mathcal{R}}$ and contains many rational points, then it is dense in an infinite semialgebraic set. Moreover, it contains an infinite set which is ${emptyset}$-definable in $langle overline{mathbb{R}}, P
angle$, where $overline{mathbb{R}}$ is the real field.
Along the way we introduce the notion of the ‘algebraic trace part’ $X^{{, alg}}_t$ of any set $Xsubseteq mathbb{R}^n$, and we show that if X is definable in an o-minimal structure, then $X^{{, alg}}_t$ coincides with the usual algebraic part of X.
The Pila–Wilkie theorem states that if a set $Xsubseteq mathbb{R}^n$ is definable in an o-minimal structure $mathcal{R}$ and contains ‘many’ rational points, then it contains an infinite semialgebraic set. In this paper, we extend this theorem to an expansion $widetilde{mathcal{R}}=langle {mathcal{R}}, P
angle$ of ${mathcal{R}}$ by a dense set P, which is either an elementary substructure of ${mathcal{R}}$, or it is $mathrm{dcl}$-independent, as follows. If X is definable in $widetilde{mathcal{R}}$ and contains many rational points, then it is dense in an infinite semialgebraic set. Moreover, it contains an infinite set which is ${emptyset}$-definable in $langle overline{mathbb{R}}, P
angle$, where $overline{mathbb{R}}$ is the real field.
Along the way we introduce the notion of the ‘algebraic trace part’ $X^{{, alg}}_t$ of any set $Xsubseteq mathbb{R}^n$, and we show that if X is definable in an o-minimal structure, then $X^{{, alg}}_t$ coincides with the usual algebraic part of X.
密集对中的小集合
影响因子：
1
作者：
P. Eleftheriou
通讯作者：
P. Eleftheriou
影响因子：
0.3
作者：
Eleftheriou, Pantelis E.;Günaydın, Ayhan;Hieronymi, Philipp
通讯作者：
Hieronymi, Philipp