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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Bibliographic Details
Published in:Foundations of computational mathematics Vol. 25; no. 2; pp. 631 - 681
Main Authors: Alberich-Carramiñana, Maria, Guàrdia, Jordi, Nart, Enric, Poteaux, Adrien, Roé, Joaquim, Weimann, Martin
Format: Journal Article
Language:English
Published: New York Springer US 01.04.2025
Springer Nature B.V
Springer Verlag
Subjects:
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
Online Access:Get full text
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Summary: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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ISSN:1615-3375
1615-3383
DOI:10.1007/s10208-024-09646-x