Polynomial factorization over Henselian fields
We present an algorithm that, given an irreducible polynomial g over a general valued field (K, v), finds the factorization of g over the Henselianization of K under certain conditions. The analysis leading to the algorithm follows the footsteps of Ore, Mac Lane, Okutsu, Montes, Vaquié and Herrera–O...
| Autores: | , , , , , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2025 |
| País: | España |
| Institución: | Universitat Politècnica de Catalunya (UPC) |
| Repositorio: | UPCommons. Portal del coneixement obert de la UPC |
| Idioma: | inglés |
| OAI Identifier: | oai:upcommons.upc.edu:2117/429176 |
| Acceso en línea: | https://hdl.handle.net/2117/429176 https://dx.doi.org/10.1007/s10208-024-09646-x |
| Access Level: | acceso abierto |
| Palabra clave: | Key polynomial Newton polygon OM-algorithm Valuation Henselian field Classificació AMS::12 Field theory and polynomials::12Y05 Computational aspects of field theory and polynomials Classificació AMS::14 Algebraic geometry::14Q Computational aspects in algebraic geometry Classificació AMS::13 Commutative rings and algebras::13P Computational aspects of commutative algebra Classificació AMS::13 Commutative rings and algebras::13A General commutative ring theory Àrees temàtiques de la UPC::Matemàtiques i estadística |
| Sumario: | We present an algorithm that, given an irreducible polynomial g over a general valued field (K, v), finds the factorization of g over the Henselianization of K under certain conditions. The analysis leading to the algorithm follows the footsteps of Ore, Mac Lane, Okutsu, Montes, Vaquié and Herrera–Olalla–Mahboub–Spivakovsky, whose work we review in our context. The correctness is based on a key new result (Theorem 4.10), exhibiting relations between generalized Newton polygons and factorization in the context of an arbitrary valuation. This allows us to develop a polynomial factorization algorithm and an irreducibility test that go beyond the classical discrete, rank-one case. These foundational results may find applications for various computational tasks involved in arithmetic of function fields, desingularization of hypersurfaces, multivariate Puiseux series or valuation theory. |
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