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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| Accès en ligne: | https://arxiv.org/abs/2503.15833 |
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| _version_ | 1866916657945903104 |
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| author | Ellenberg, Jordan S. Hardt, Will |
| author_facet | Ellenberg, Jordan S. Hardt, Will |
| contents | In a 1986 paper, Smyth proposed a conjecture about which integer-linear relations were possible among Galois-conjugate algebraic numbers. We prove this conjecture. The main tools (as Smyth already anticipated) are combinatorial rather than number-theoretic in nature. For instance, the question can be reinterpreted as a question about the possible eigenvalues of a specified linear combination of permutation matrices. What's more, we reinterpret Smyth's conjecture as a local-to-global principle for a "non-deterministic system of equations" where variables are interpreted as compactly supported K-valued random variables (for K a local or global field) rather than as elements of K. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_15833 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Smyth's conjecture and a non-deterministic Hasse principle Ellenberg, Jordan S. Hardt, Will Number Theory Combinatorics 14G12 (Primary) 05C65, 11R04 (Secondary) In a 1986 paper, Smyth proposed a conjecture about which integer-linear relations were possible among Galois-conjugate algebraic numbers. We prove this conjecture. The main tools (as Smyth already anticipated) are combinatorial rather than number-theoretic in nature. For instance, the question can be reinterpreted as a question about the possible eigenvalues of a specified linear combination of permutation matrices. What's more, we reinterpret Smyth's conjecture as a local-to-global principle for a "non-deterministic system of equations" where variables are interpreted as compactly supported K-valued random variables (for K a local or global field) rather than as elements of K. |
| title | Smyth's conjecture and a non-deterministic Hasse principle |
| topic | Number Theory Combinatorics 14G12 (Primary) 05C65, 11R04 (Secondary) |
| url | https://arxiv.org/abs/2503.15833 |