有理点存在的有效性是不可确定的

IF 0.6 3区数学 Q3 MATHEMATICS

Journal of Number Theory Pub Date : 2025-05-06 DOI:10.1016/j.jnt.2025.01.023

Natalia Garcia-Fritz , Hector Pasten , Xavier Vidaux

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引用次数: 0

摘要

在Q上的希尔伯特第十问题的类似问题要求一种算法来确定该域上代数变异上有理点的存在性。这仍然是数论中不可判定性领域的主要开放问题之一。除了有理点的存在性之外，对有效性问题也有相当大的兴趣：人们问所寻找的有理点是否满足确定的高度界限，通常用定义所考虑的代数变量的方程的系数的高度来表示。我们证明，事实上，希尔伯特的第十问题在Q上具有（有限多个）高度比较条件是不可判定的。

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Effectivity for existence of rational points is undecidable

The analogue of Hilbert's tenth problem over

Q

asks for an algorithm to decide the existence of rational points on algebraic varieties over this field. This remains as one of the main open problems in the area of undecidability in number theory. Besides the existence of rational points, there is also considerable interest in the problem of effectivity: one asks whether the sought rational points satisfy determined height bounds, often expressed in terms of the height of the coefficients of the equations defining the algebraic varieties under consideration. We show that, in fact, Hilbert's tenth problem over

Q

with (finitely many) height comparison conditions is undecidable.

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来源期刊

Journal of Number Theory 数学-数学

CiteScore

1.30

自引率

14.30%

发文量

122

审稿时长

16 weeks

期刊介绍： The Journal of Number Theory (JNT) features selected research articles that represent the broad spectrum of interest in contemporary number theory and allied areas. A valuable resource for mathematicians, the journal provides an international forum for the publication of original research in this field. The Journal of Number Theory is encouraging submissions of quality, long articles where most or all of the technical details are included. The journal now considers and welcomes also papers in Computational Number Theory. Starting in May 2019, JNT will have a new format with 3 sections: JNT Prime targets (possibly very long with complete proofs) high impact papers. Articles published in this section will be granted 1 year promotional open access. JNT General Section is for shorter papers. We particularly encourage submission from junior researchers. Every attempt will be made to expedite the review process for such submissions. Computational JNT . This section aims to provide a forum to disseminate contributions which make significant use of computer calculations to derive novel number theoretic results. There will be an online repository where supplementary codes and data can be stored.