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Encoding Algebraic Power Series

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Abstract

The division algorithm for ideals of algebraic power series satisfying Hironaka’s box condition is shown to be finite when expressed suitably in terms of the defining polynomial codes of the series. In particular, the codes of the reduced standard basis of the ideal can be constructed effectively.

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Acknowledgements

The authors are very indebted to T. Mora, W. Seiler, G.-M. Greuel, G. Rond and O. Villamayor for many valuable comments and helpful suggestions during the preparation of this article. W. Seiler pointed out an inaccuracy in an earlier version of the manuscript, indicated the relation of echelons and Janet bases with his notion of \(\delta \)-regularity and Pommaret division, and provided several important references. We are also grateful to two anonymous referees for valuable suggestions of how to improve the exposition. Part of the work on the article has been done during visits of the first two authors to the University of Vienna and the Erwin Schrödinger Institute, of the third author to the Universities of Seville and Complutense de Madrid, and during the special semester on Artin approximation within the Chaire Jean Morlet of the third author at CIRM, Luminy.

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Authors and Affiliations

  1. Departamento de Álgebra, Universidad Complutense de Madrid, Madrid, Spain

    M. E. Alonso

  2. Departamento de Álgebra, Universidad de Sevilla, Seville, Spain

    F. J. Castro-Jiménez

  3. Fakultät für Mathematik, Universität Wien, Vienna, Austria

    H. Hauser

  4. Institut für Mathematik, Universität Innsbruck, Innsbruck, Austria

    H. Hauser

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  1. M. E. Alonso
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  2. F. J. Castro-Jiménez
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  3. H. Hauser
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Correspondence to M. E. Alonso.

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Communicated by Joseph Landsberg.

M.E.A. acknowledges support from MEC, 2011-22435, and UCM, Grupo 910444; F.J.C.-J. from MTM2010-19336, MTM2013-40455-P and Feder; H.H. from the Austrian Science Fund FWF through projects P-21461 and P-25652. Part of the work was done during the special semester on Artin approximation within the Chaire Jean Morlet of the third author at CIRM, Luminy.

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Alonso, M.E., Castro-Jiménez, F.J. & Hauser, H. Encoding Algebraic Power Series. Found Comput Math 18, 789–833 (2018). https://doi.org/10.1007/s10208-017-9354-z

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