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Effective Power Series Computations

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Abstract

Let K be an effective field of characteristic zero. An effective tribe is a subset of \(K [[z_1, z_2, \ldots ]] = K \cup K {[}[z_1]] \cup K [[z_1, z_2]] \cup \cdots \) that is effectively stable under the K-algebra operations, restricted division, composition, the implicit function theorem, as well as restricted monomial transformations with arbitrary rational exponents. Given an effective tribe with an effective zero test, we will prove that an effective version of the Weierstrass division theorem holds inside the tribe and that this can be used for the computation of standard bases.

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Acknowledgements

The author wishes to express his gratitude to the two referees for their careful reading and helpful comments, as well as to Alin Bostan and Lou van den Dries for historical references.

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Authors and Affiliations

  1. CNRS, LIX, École Polytechnique, 91128, Palaiseau Cedex, France

    Joris van der Hoeven

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  1. Joris van der Hoeven
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Correspondence to Joris van der Hoeven.

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Communicated by Joseph M. Landsberg.

This work has been supported by the ANR-10-BLAN 0109 LEDA Project.

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van der Hoeven, J. Effective Power Series Computations. Found Comput Math 19, 623–651 (2019). https://doi.org/10.1007/s10208-018-9391-2

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  • DOI: https://doi.org/10.1007/s10208-018-9391-2

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