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Computing the Homology of Semialgebraic Sets. II: General Formulas

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Abstract

We describe and analyze a numerical algorithm for computing the homology (Betti numbers and torsion coefficients) of semialgebraic sets given by Boolean formulas. The algorithm works in weak exponential time. This means that outside a subset of data having exponentially small measure, the cost of the algorithm is single exponential in the size of the data. This extends the work in Part I to arbitrary semialgebraic sets. All previous algorithms proposed for this problem have doubly exponential complexity.

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Notes

  1. Claims in [1, 2] regarding the parallelization of the computation of torsion coefficients were inaccurate. As of today, the main difficulty lies in showing the existence of efficient parallel algorithms for the computation of the Smith Normal Form of integer matrices. See [17, p. 160-161] for more details.

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Acknowledgements

We are grateful to Nicolai Vorobjov who pointed us to (what we call here) Gabrielov–Vorobjov approximations.

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Authors and Affiliations

  1. Institut für Mathematik, Technische Universität Berlin, Berlin, Germany

    Peter Bürgisser

  2. Department of Mathematics, City University of Hong Kong, Hong Kong, China

    Felipe Cucker

  3. Inria Paris and IMJ-PRG, OURAGAN Team, Sorbonne Université, Paris, France

    Josué Tonelli-Cueto

Authors
  1. Peter Bürgisser
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  2. Felipe Cucker
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  3. Josué Tonelli-Cueto
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Correspondence to Josué Tonelli-Cueto.

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Communicated by Shmuel Weinberger.

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This work was supported by the Einstein Foundation Berlin. Peter Bürgisser: Partially funded by the European Research Council (ERC) under the European’s Horizon 2020 research and innovation programme (Grant Agreement No 787840). Felipe Cucker: Partially supported by a GRF grant from the Research Grants Council of the Hong Kong SAR (Project Number CityU 11302418). Josué Tonelli-Cueto: Partially supported by the Fondation Sciences Mathématiques de Paris, the ANR JCJC GALOP (ANR-17-CE40-0009), the PGMO grant ALMA, and the PHC GRAPE.

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Bürgisser, P., Cucker, F. & Tonelli-Cueto, J. Computing the Homology of Semialgebraic Sets. II: General Formulas. Found Comput Math 21, 1279–1316 (2021). https://doi.org/10.1007/s10208-020-09483-8

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  • DOI: https://doi.org/10.1007/s10208-020-09483-8

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