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Polynomial Factorization Over Henselian Fields

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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.

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Acknowledgements

We warmly thank the referees, whose enlightening comments led us to improve the presentation, and Josnei Novacoski for sharing his insights about the content of Sect. 4.

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Authors and Affiliations

  1. 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, Av. Diagonal, 647, 08028, Barcelona, Catalonia, Spain

    Maria Alberich-Carramiñana

  2. Departament de Matemàtiques, Escola Politècnica Superior d’Enginyeria de Vilanova i la Geltrú, Av. Víctor Balaguer s/n., 08800, Vilanova i la Geltrú, Catalonia, Spain

    Jordi Guàrdia

  3. Departament de Matemàtiques, Universitat Autònoma de Barcelona, Edifici C, 08193, Bellaterra, Barcelona, Catalonia, Spain

    Enric Nart & Joaquim Roé

  4. Université de Lille, CNRS,Centrale Lille, UMR 9189 CRIStAL, 59000, Lilles, France

    Adrien Poteaux

  5. LMNO, UMR 6139, Université de Caen-Normandie, 14032, Caen, France

    Martin Weimann

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  1. Maria Alberich-Carramiñana
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  2. Jordi Guàrdia
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  3. Enric Nart
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  4. Adrien Poteaux
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  5. Joaquim Roé
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Correspondence to Martin Weimann.

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Communicated by Christophe Ritzenthaler.

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Alberich-Carramiñana, M., Guàrdia, J., Nart, E. et al. Polynomial Factorization Over Henselian Fields. Found Comput Math (2024). https://doi.org/10.1007/s10208-024-09646-x

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