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 Herre...

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Vydané v:	Foundations of computational mathematics Ročník 25; číslo 2; s. 631 - 681
Hlavní autori:	Alberich-Carramiñana, Maria, Guàrdia, Jordi, Nart, Enric, Poteaux, Adrien, Roé, Joaquim, Weimann, Martin
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	New York Springer US 01.04.2025 Springer Nature B.V Springer Verlag
Predmet:	Algorithms Applications of Mathematics Computer Science Context Economics Factorization Hyperspaces Linear and Multilinear Algebras Math Applications in Computer Science Mathematics Mathematics and Statistics Matrix Theory Numerical Analysis Polynomials Symbolic Computation Newton polygon OM-algorithm Valuation Henselian field 12Y05 Key polynomial 13P05 14Q15 13A18
ISSN:	1615-3375, 1615-3383
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Abstract	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.
AbstractList	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. 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. Given a valued field $(K,v)$ and an irreducible polynomial $g\in K[x]$, we survey the ideas of Ore, Maclane, Okutsu, Montes, Vaqui\'e and Herrera-Olalla-Mahboub-Spivakovsky, leading (under certain conditions) to an algorithm to find the factorization of $g$ over a henselization of $(K,v)$.
Author	Roé, Joaquim Poteaux, Adrien Weimann, Martin Nart, Enric Alberich-Carramiñana, Maria Guàrdia, Jordi
Author_xml	– sequence: 1 givenname: Maria surname: Alberich-Carramiñana fullname: Alberich-Carramiñana, Maria organization: Institut de Robòtica i Informàtica Industrial (IRI, CSIC-UPC), Institut de Matemátiques de la UPC-BarcelonaTech (IMTech) and Departament de Matemàtiques, Universitat Politècnica de Catalunya. BarcelonaTech – sequence: 2 givenname: Jordi surname: Guàrdia fullname: Guàrdia, Jordi organization: Departament de Matemàtiques, Escola Politècnica Superior d’Enginyeria de Vilanova i la Geltrú – sequence: 3 givenname: Enric surname: Nart fullname: Nart, Enric organization: Departament de Matemàtiques, Universitat Autònoma de Barcelona – sequence: 4 givenname: Adrien surname: Poteaux fullname: Poteaux, Adrien organization: Université de Lille, CNRS,Centrale Lille, UMR 9189 CRIStAL – sequence: 5 givenname: Joaquim surname: Roé fullname: Roé, Joaquim organization: Departament de Matemàtiques, Universitat Autònoma de Barcelona – sequence: 6 givenname: Martin surname: Weimann fullname: Weimann, Martin email: martin.weimann@unicaen.fr organization: LMNO, UMR 6139, Université de Caen-Normandie
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Snippet	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... 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... Given a valued field $(K,v)$ and an irreducible polynomial $g\in K[x]$, we survey the ideas of Ore, Maclane, Okutsu, Montes, Vaqui\'e and...
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SubjectTerms	Algorithms Applications of Mathematics Computer Science Context Economics Factorization Hyperspaces Linear and Multilinear Algebras Math Applications in Computer Science Mathematics Mathematics and Statistics Matrix Theory Numerical Analysis Polynomials Symbolic Computation
Title	Polynomial Factorization Over Henselian Fields
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