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Polynomial-Time Amoeba Neighborhood Membership and Faster Localized Solving

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Topological and Statistical Methods for Complex Data

Part of the book series: Mathematics and Visualization ((MATHVISUAL))

Abstract

We derive efficient algorithms for coarse approximation of complex algebraic hypersurfaces, useful for estimating the distance between an input polynomial zero set and a given query point. Our methods work best on sparse polynomials of high degree (in any number of variables) but are nevertheless completely general. The underlying ideas, which we take the time to describe without an excess of algebraic geometry terminology, come from tropical geometry. We then apply our methods to finding roots of n × n systems near a given query point, thereby reducing a hard algebraic problem to high-precision linear optimization. We prove new upper and lower complexity estimates along the way.

Dedicated to Tien-Yien Li, in honor of his birthday.

Partially supported by NSF REU grant DMS-1156589.

Partially supported by NSF REU grant DMS-1156589.

Work supported in part by the German Research Foundation (DFG).

Partially supported by NSF MCS grant DMS-0915245.

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Notes

  1. 1.

    A hole of a subset \(S\! \subseteq \! \mathbb{R}^{n}\) is simply a bounded connected component of the complement \(\mathbb{R}^{n}\setminus S\).

  2. 2.

    That is, smallest convex set containing…

  3. 3.

    The cell looks hexagonal because it has a pair of vertices too close to distinguish visually.

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Author information

Authors and Affiliations

  1. Mathematics Department, University of Mississippi, Hume Hall 305, 1848, Oxford, MS, 38677-1848, USA

    Eleanor Anthony

  2. Mathematics Department, 640 North College Avenue, Claremont, CA, 91711, USA

    Sheridan Grant

  3. Fakultät für Mathematik, Technische Universität München, 80290, München, Germany

    Peter Gritzmann

  4. Mathematics Department, Texas A&M University, TAMU 3368, College Station, TX, 77843-3368, USA

    J. Maurice Rojas

Authors
  1. Eleanor Anthony
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  2. Sheridan Grant
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  3. Peter Gritzmann
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  4. J. Maurice Rojas
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Correspondence to J. Maurice Rojas .

Editor information

Editors and Affiliations

  1. Sandia National Laboratories, MS 9152, Livermore, California, USA

    Janine Bennett

  2. CEA CESTA, CS 60001, Le Barp CEDEX, France

    Fabien Vivodtzev

  3. School of Computing and SCI Institute, University of Utah, Salt Lake City, Utah, USA

    Valerio Pascucci

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Anthony, E., Grant, S., Gritzmann, P., Rojas, J.M. (2015). Polynomial-Time Amoeba Neighborhood Membership and Faster Localized Solving. In: Bennett, J., Vivodtzev, F., Pascucci, V. (eds) Topological and Statistical Methods for Complex Data. Mathematics and Visualization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-44900-4_15

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