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Variations on inversion theorems for Newton–Puiseux series

Mathematische Annalen volume 368, pages 1359–1397 (2017)Cite this article

Abstract

Let f(x, y) be an irreducible formal power series without constant term, over an algebraically closed field of characteristic zero. One may solve the equation \(f(x,y)=0\) by choosing either x or y as independent variable, getting two finite sets of Newton–Puiseux series. In 1967 and 1968 respectively, Abhyankar and Zariski published proofs of an inversion theorem, expressing the characteristic exponents of one set of series in terms of those of the other set. In fact, a more general theorem, stated by Halphen in 1876 and proved by Stolz in 1879, relates also the coefficients of the characteristic terms of both sets of series. This theorem seems to have been completely forgotten. We give two new proofs of it and we generalize it to a theorem concerning irreducible series with an arbitrary number of variables.

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Acknowledgements

This research was partially supported by the French grants ANR-12-JS01-0002-01 SUSI, Labex CEMPI ANR-11-LABX-0007-01 and the Spanish grants MTM2012-36917-C03-0, MTM2013-45710-C2-2-P, MTM2016-76868-C2-1-P, MTM2016-80659-P. We are grateful to Herwig Hauser and Hana Kováčová for their translation of parts of Stolz’ paper. The third-named author is grateful to Mickaël Matusinski for the opportunity to explain our first proof of the Halphen–Stolz theorem at the Geometry seminar of the University of Bordeaux. We thank him and the anonymous referee for their remarks on a previous version of this article, which allowed us to improve our presentation. We also thank Antonio Campillo for the information he sent us about the uses of Abhyankar–Zariski inversion.

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Authors and Affiliations

  1. Departamento de Matemáticas, Estadística e I.O. Sección de Matemáticas, Universidad de La Laguna (ULL), Apartado de Correos 456. 38200 La Laguna, Tenerife, Spain

    Evelia Rosa García Barroso

  2. Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), Departamento de Álgebra, Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, Plaza de las Ciencias 3, Madrid, 28070, Spain

    Pedro Daniel González Pérez

  3. Univ. Lille, UMR 8524, Laboratoire Paul Painlevé, 59000, Lille, France

    Patrick Popescu-Pampu

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  1. Evelia Rosa García Barroso
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  2. Pedro Daniel González Pérez
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  3. Patrick Popescu-Pampu
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Correspondence to Patrick Popescu-Pampu.

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García Barroso, E.R., González Pérez, P.D. & Popescu-Pampu, P. Variations on inversion theorems for Newton–Puiseux series. Math. Ann. 368, 1359–1397 (2017). https://doi.org/10.1007/s00208-016-1503-1

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  • DOI: https://doi.org/10.1007/s00208-016-1503-1

Keywords

  • Branch
  • Characteristic exponents
  • Plane curve singularities
  • Hypersurface singularities
  • Quasi-ordinary series
  • Lagrange inversion
  • Newton-Puiseux series

Mathematics Subject Classification

  • Primary 14B05
  • 14H20
  • 32S25