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Computing the Homology of Real Projective Sets

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Abstract

We describe and analyze a numerical algorithm for computing the homology (Betti numbers and torsion coefficients) of real projective varieties. Here numerical means that the algorithm is numerically stable (in a sense to be made precise). Its cost depends on the condition of the input as well as on its size and is singly exponential in the number of variables (the dimension of the ambient space) and polynomial in the condition and the degrees of the defining polynomials. In addition, we show that outside of an exceptional set of measure exponentially small in the size of the data, the algorithm takes exponential time.

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Acknowledgements

We are grateful to Peter Bürgisser who suggested the topic of this paper to us and to the Simons Institute for receiving us in the Fall of 2014, which was where and when the suggestion was made. We also owe an anonymous referee for his very precise and enlightening comments.

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Authors and Affiliations

  1. Department of Mathematics, City University of Hong Kong, Kowloon Tong, Hong Kong

    Felipe Cucker

  2. Departamento de Matemática & IMAS, Univ. de Buenos Aires & CONICET, Buenos Aires, Argentina

    Teresa Krick

  3. Department of Mathematics, City College and the Graduate Center of CUNY, New York, NY, USA

    Michael Shub

Authors
  1. Felipe Cucker
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  2. Teresa Krick
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  3. Michael Shub
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Corresponding author

Correspondence to Teresa Krick.

Additional information

Communicated by Shmuel Weinberger.

Felipe Cucker partially supported by a GRF Grant from the Research Grants Council of the Hong Kong SAR (Project Number CityU 11310716). Teresa Krick partially supported by Grants BID-PICT-2013-0294, UBACyT-2014-2017-20020130100143BA and PIP-CONICET 2014-2016-11220130100073CO.

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Cucker, F., Krick, T. & Shub, M. Computing the Homology of Real Projective Sets. Found Comput Math 18, 929–970 (2018). https://doi.org/10.1007/s10208-017-9358-8

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