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Newton Polygon

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Handbook of Geometry and Topology of Singularities V: Foliations

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Abstract

This paper is an introduction to the use of techniques originally developed by Newton to study local solutions of algebraic and ordinary differential equations. We first study the application of Newton’s polygon to algebraic equations with coefficients in a valued field, showing the limitations of its use for valuations of rank greater than one. The Theorem of Kaplansky is the key to have explicit solutions for the rank one case. For differential equations, the lack of a Differential Kaplansky’s theorem, is an obstacle to have a general theory, but we study ordinary differential equations with generalized series with exponents in subgroups of \(\mathbb {R}\) as coefficients. In booth cases we pointed up the possibility of the use valuations as solution of equations. The author wishes to thank to the referee for his (many) precise and detailed corrections that have improved this paper.

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Notes

  1. 1.

    Generalized series were introduced by Hahn [27], and in a slightly stronger form by MacLane [41] who called them simply formal series. Some authors call them Hahn’s series, and for some others as Grigoriev and Singer [26] generalized series are the series of Example 8.2.6\((4)\).

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Authors and Affiliations

  1. Centro “Tordesillas” de Relaciones con Iberoamérica (CTRI), Universidad de Valladolid, Tordesillas, Spain

    José Manuel Aroca

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  1. José Manuel Aroca
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Correspondence to José Manuel Aroca .

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Editors and Affiliations

  1. Departamento de Algebra, Análisis Matemático, Geometría y Topología, Universidad de Valladolid, Valladolid, Spain

    Felipe Cano

  2. Instituto de Matemáticas, Unidad Cuernavaca, Universidad Nacional Autónoma de México, Cuernavaca, Mexico City, Mexico

    José Luis Cisneros-Molina

  3. Centre de mathématiques et informatique, Université d’Aix-Marseille, Marseille, France

    Lê Dũng Tráng

  4. Instituto de Matemáticas, Unidad Cuernavaca, Universidad Nacional Autónoma de México, Cuernavaca, Mexico City, Mexico

    José Seade

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Aroca, J.M. (2024). Newton Polygon. In: Cano, F., Cisneros-Molina, J.L., Dũng Tráng, L., Seade, J. (eds) Handbook of Geometry and Topology of Singularities V: Foliations. Springer, Cham. https://doi.org/10.1007/978-3-031-52481-3_8

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