房子的诺斯科特数和韦尔高度
Northcott numbers for the house and the Weil height
摘要：
对于代数数$α$和$γ\in \mathbb{R}$，$h(α)$是（对数）Weil高度，而$h_γ(α)=(\mathrm{deg}α)^γh(α)$是$α$的$γ$加权（对数）Weil高度.令$f:\overline{\mathbb{Q}}\to [0,\infty)$是代数数$\overline{\mathbb{Q}}$上的函数，令$S\subset \overline{\mathbb{Q}}$。$S$的Northcott数$\mathcal{N}_f(S)$，相对于$f$，是所有$X\geq 0$的下确界，使得$\{α\in S; f(α)< X\}$是无限的。本文研究了房子的$\overline{\mathbb{Q}}$的子环的Northcott数集$\mathcal{N}_f(\mathcal{O})$、Weil高度和$γ$加权Weil高度.证明了：（1）每个$t\geq 1$是域w.r.t.的整数环的Northcott数;房子(2)对于每个PDMATHBNB，在PDMATHBNC w.r.t.中存在具有Northcott数的字段。Weil高度$h(\cdot)$。(3)对于所有的$0\leq γ\leq 1$和$γ'<γ$，存在一个带有$\mathcal{N}_{h_{γ'}}(K)=0$和$\mathcal{N}_{h_γ}(K)=\infty$的字段$K$。对于$(1)$，我们提供了满足Julia Robinon性质（JR）的类似物的例子，满足Vidaux和Videla隔离性质的类似物的例子，以及不满足这些的例子。项目$(2)$涉及Vidaux和Videla提出的一个问题，因为它通过Julia Robinson数与可判定性理论直接联系。第（3）项是对已知事实的有力概括，即存在满足莱默猜想的域，但它们不是Bogomolov意义上的Bogomolov和Zannier。
英文摘要：
For an algebraic number $α$ and $γ\in \mathbb{R}$, $h(α)$ be the (logarithmic) Weil height, and $h_γ(α)=(\mathrm{deg}α)^γh(α)$ be the $γ$-weighted (logarithmic) Weil height of $α$. Let $f:\overline{\mathbb{Q}}\to [0,\infty)$ be a function on the algebraic numbers $\overline{\mathbb{Q}}$, and let $S\subset \overline{\mathbb{Q}}$. The Northcott number $\mathcal{N}_f(S)$ of $S$, with respect to $f$, is the infimum of all $X\geq 0$ such that $\{α\in S; f(α)< X\}$ is infinite. This paper studies the set of Northcott numbers $\mathcal{N}_f(\mathcal{O})$ for subrings of $\overline{\mathbb{Q}}$ for the house, the Weil height, and the $γ$-weighted Weil height. We show: (1) Every $t\geq 1$ is the Northcott number of a ring of integers of a field w.r.t. the house. (2) For each $t\geq 0$ there exists a field with Northcott number in $ [t,2t]$ w.r.t. the Weil height $h(\cdot)$. (3) For all $0\leq γ\leq 1$ and $γ'<γ$ there exists a field $K$ with $\mathcal{N}_{h_{γ'}}(K)=0$ and $\mathcal{N}_{h_γ}(K)=\infty$. For $(1)$ we provide examples that satisfy an analogue of Julia Robinon's property (JR), examples that satisfy an analogue of Vidaux and Videla's isolation property, and examples that satisfy neither of those. Item $(2)$ concerns a question raised by Vidaux and Videla due to its direct link with decidability theory via the Julia Robinson number. Item (3) is a strong generalisation of the known fact that there are fields that satisfy the Lehmer conjecture but which are not Bogomolov in the sense of Bombieri and Zannier.
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