论文标题
$ \ mathbb {q} $的代数扩展中的不可限权的拓扑方法
A topological approach to undefinability in algebraic extensions of $\mathbb{Q}$
论文作者
论文摘要
对于任何子集$ z \ subseteq \ mathbb {q} $，请考虑子场的集合$ s_z $ $ l \ subseteq \ overline {\ mathbb {q}} $，该{\ mathbb {q}} $包含一个co-Infinite $ c \ subseteq l $，在$ c \ c \ c cap cop cap co.c \ cap cap cap cap cap =将自然拓扑放在集合$ \ text {sub}（\ overline {\ mathbb {q}}}）$ + overline {\ mathbb {q}} $的子场的$中，我们表明，如果$ z $在$ \ mathbb {q} $中不是很薄的，则是$ \ s_ $ s_z $ s_eger insmeager in $ \ text {sub}（\ overline {\ mathbb {q}}}）$。在这里，就算术几何和拓扑而言，薄而微薄的意思是“小”。例如，这意味着只有一组微薄的字段$ l $具有代数整数环$ \ Mathcal {o} _l $的属性，在$ l $中是普遍定义的。主要工具是希尔伯特的不可约性定理和生存定义的新常规形式定理。可能具有独立利益的正常形式定理大致说，每一个$ \ cy的每一个可定义的代数扩展子集的$ \ mathbb Q $都是单点和由绝对是不可约多综合族所定义的单点和预测的有限结合。
For any subset $Z \subseteq \mathbb{Q}$, consider the set $S_Z$ of subfields $L\subseteq \overline{\mathbb{Q}}$ which contain a co-infinite subset $C \subseteq L$ that is universally definable in $L$ such that $C \cap \mathbb{Q}=Z$. Placing a natural topology on the set $\text{Sub}(\overline{\mathbb{Q}})$ of subfields of $\overline{\mathbb{Q}}$, we show that if $Z$ is not thin in $\mathbb{Q}$, then $S_Z$ is meager in $\text{Sub}(\overline{\mathbb{Q}})$. Here, thin and meager both mean "small", in terms of arithmetic geometry and topology, respectively. For example, this implies that only a meager set of fields $L$ have the property that the ring of algebraic integers $\mathcal{O}_L$ is universally definable in $L$. The main tools are Hilbert's Irreducibility Theorem and a new normal form theorem for existential definitions. The normal form theorem, which may be of independent interest, says roughly that every $\exists$-definable subset of an algebraic extension of $\mathbb Q$ is a finite union of single points and projections of hypersurfaces defined by absolutely irreducible polynomials.
