Defining $\mathbb{Z}$ in $\mathbb{Q}$
| Autor: | Koenigsmann, Jochen |
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| Rok vydání: | 2010 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | We show that ${\mathbb Z}$ is definable in ${\mathbb Q}$ by a universal first-order formula in the language of rings. We also present an $\forall\exists$-formula for ${\mathbb Z}$ in ${\mathbb Q}$ with just one universal quantifier. We exhibit new diophantine subsets of ${\mathbb Q}$ like the complement of the image of the norm map under a quadratic extension, and we give an elementary proof of the fact that the set of non-squares is diophantine. Finally, we show that there is no existential formula for ${\mathbb Z}$ in ${\mathbb Q}$, provided one assumes a strong variant of the Bombieri-Lang Conjecture for varieties over ${\mathbb Q}$ with many ${\mathbb Q}$-rational points. Comment: New shorter proofs of Proposition 16 (a) and Corollary 23, improved presentation and many tiny corrections |
| Databáze: | arXiv |
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