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Preprint Number 2519
2519. Natalia Garcia-Fritz, Hector Pasten, Xavier Vidaux Effectivity for existence of rational points is undecidable E-mail: Submission date: 3 November 2023 Abstract: The analogue of Hilbert's tenth problem over ℚ asks for an algorithm to decide the existence of rational points in 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. Mathematics Subject Classification: Primary: 11U05, Secondary: 11C08, 11G50 Keywords and phrases: |
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