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The minimal cone of an algebraic Laurent series

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Abstract

We study the algebraic closure of \(\mathbb {K}(\!(x)\!)\), the field of power series in several indeterminates over a field \(\mathbb {K}\). In characteristic zero we show that the elements algebraic over \(\mathbb {K}(\!(x)\!)\) can be expressed as Puiseux series such that the convex hull of its support is essentially a polyhedral rational cone, strengthening the known results. In positive characteristic we construct algebraic closed fields containing the field of power series and we give examples showing that the results proved in characteristic zero are no longer valid in positive characteristic.

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Authors and Affiliations

  1. Instituto de Matemáticas, Universidad Nacional Autónoma de México (UNAM), Coyoacan, Mexico

    Fuensanta Aroca & Julie Decaup

  2. Université Publique, CNRS, Centrale Marseille, I2M, UMR 7373, 13453, Marseille, France

    Guillaume Rond

Authors
  1. Fuensanta Aroca
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  2. Julie Decaup
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  3. Guillaume Rond
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Correspondence to Guillaume Rond.

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Communicated by Jean-Yves Welschinger.

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This work has been partially supported by ECOS Project M14M03, and by PAPIIT IN108216 and IN108320. The third author is deeply grateful to the UMI LASOL of CNRS where this project has been carried out.

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Aroca, F., Decaup, J. & Rond, G. The minimal cone of an algebraic Laurent series. Math. Ann. 382, 1745–1773 (2022). https://doi.org/10.1007/s00208-021-02338-9

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  • DOI: https://doi.org/10.1007/s00208-021-02338-9

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