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Cell decomposition of almost smooth real algebraic surfaces

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Abstract

Let Z be a two dimensional irreducible complex component of the solution set of a system of polynomial equations with real coefficients in N complex variables. This work presents a new numerical algorithm, based on homotopy continuation methods, that begins with a numerical witness set for Z and produces a decomposition into 2-cells of any almost smooth real algebraic surface contained in Z. Each 2-cell (a face) has a generic interior point and a boundary consisting of 1-cells (edges). Similarly, the 1-cells have a generic interior point and a vertex at each end. Each 1-cell and each 2-cell has an associated homotopy for moving the generic interior point to any other point in the interior of the cell, defining an invertible map from the parameter space of the homotopy to the cell. This work draws on previous results for the curve case. Once the cell decomposition is in hand, one can sample the 2-cells and 1-cells to any resolution, limited only by the computational resources available.

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Author information

Authors and Affiliations

  1. College of Computing and Digital Media, DePaul University, 243 S Wabash, Chicago, IL, 60604, USA

    Gian Mario Besana

  2. Department of Mathematics, KTH, 10044, Stockholm, Sweden

    Sandra Di Rocco

  3. Department of Mathematics, North Carolina State University, Raleigh, NC, 27695, USA

    Jonathan D. Hauenstein

  4. Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, IN, 46556, USA

    Andrew J. Sommese

  5. General Motors Research and Development, 480-106-224, 30500 Mound Road, Warren, MI, 48090, USA

    Charles W. Wampler

Authors
  1. Gian Mario Besana
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  2. Sandra Di Rocco
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  3. Jonathan D. Hauenstein
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  4. Andrew J. Sommese
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  5. Charles W. Wampler
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Corresponding author

Correspondence to Charles W. Wampler.

Additional information

S. Di Rocco was supported by the Mittag-Leffler Institute, the University of Notre Dame, and VR grant NT:2010-5563. J. D. Hauenstein was supported by the Mittag-Leffler Institute and NSF grants DMS-0915211 and DMS-1114336. A. J. Sommese was supported by the Mittag-Leffler Institute, the Duncan Chair of the University of Notre Dame, and NSF grant DMS-0712910. C. W. Wampler was supported by the Mittag-Leffler Institute and NSF grant DMS-0712910.

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Besana, G.M., Di Rocco, S., Hauenstein, J.D. et al. Cell decomposition of almost smooth real algebraic surfaces. Numer Algor 63, 645–678 (2013). https://doi.org/10.1007/s11075-012-9646-y

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