Defining ℤ in ℚ
We show that Z is definable in Q by a universal first-order formula in the language of rings. We also present an ∀∃-formula for Z in Q with just one universal quantifier. We exhibit new diophantine subsets of Q like the complement of the image of the norm map under a quadratic extension, and we give...
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| Format: | Journal article |
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Princeton University, Department of Mathematics
2016
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| _version_ | 1797092388753113088 |
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| author | Koenigsmann, J |
| author_facet | Koenigsmann, J |
| author_sort | Koenigsmann, J |
| collection | OXFORD |
| description | We show that Z is definable in Q by a universal first-order formula in the language of rings. We also present an ∀∃-formula for Z in Q with just one universal quantifier. We exhibit new diophantine subsets of Q like the complement of the image of the norm map under a quadratic extension, and we give an elementary proof for the fact that the set of non-squares is diophantine. Finally, we show that there is no existential formula for Z in Q, provided one assumes a strong variant of the Bombieri-Lang Conjecture for varieties over Q with many Q-rational points. |
| first_indexed | 2024-03-07T03:45:18Z |
| format | Journal article |
| id | oxford-uuid:bf44cb00-150e-4e4d-80c5-a123ae5cc21d |
| institution | University of Oxford |
| last_indexed | 2024-03-07T03:45:18Z |
| publishDate | 2016 |
| publisher | Princeton University, Department of Mathematics |
| record_format | dspace |
| spelling | oxford-uuid:bf44cb00-150e-4e4d-80c5-a123ae5cc21d2022-03-27T05:46:06ZDefining ℤ in ℚJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:bf44cb00-150e-4e4d-80c5-a123ae5cc21dSymplectic Elements at OxfordPrinceton University, Department of Mathematics2016Koenigsmann, JWe show that Z is definable in Q by a universal first-order formula in the language of rings. We also present an ∀∃-formula for Z in Q with just one universal quantifier. We exhibit new diophantine subsets of Q like the complement of the image of the norm map under a quadratic extension, and we give an elementary proof for the fact that the set of non-squares is diophantine. Finally, we show that there is no existential formula for Z in Q, provided one assumes a strong variant of the Bombieri-Lang Conjecture for varieties over Q with many Q-rational points. |
| spellingShingle | Koenigsmann, J Defining ℤ in ℚ |
| title | Defining ℤ in ℚ |
| title_full | Defining ℤ in ℚ |
| title_fullStr | Defining ℤ in ℚ |
| title_full_unstemmed | Defining ℤ in ℚ |
| title_short | Defining ℤ in ℚ |
| title_sort | defining z in q |
| work_keys_str_mv | AT koenigsmannj definingzinq |