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An Efficient Algorithm for the Sign Condition Problem in the Semi-algebraic Context

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Advances in Geometric Modeling and Processing (GMP 2010)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6130))

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Abstract

We study algebraic complexity of the sign condition problem for any given family of polynomials. Essentially, the problem consists in determining the sign condition satisfied by a fixed family of polynomials at a query point, performing as little arithmetic operations as possible. After defining precisely the sign condition and the point location problems, we introduce a method called the dialytic method to solve the first problem efficiently. This method involves a linearization of the original polynomials and provides the best known algorithm to solve the sign condition problem. Moreover, we prove a lower bound showing that the dialytic method is almost optimal. Finally, we extend our method to the point location problem.

The dialytic method solves (non-uniformly) the sign condition problem for a family of s polynomials in R[X 1,...,X n ] given by an arithmetic circuit \(\Gamma_{\mathcal F}\) of non-scalar complexity L performing \({\mathcal O}((L+n)^5\log(s))\) arithmetic operations.

If the polynomials are given in dense representation and d is a bound for their degrees, the complexity of our method is \({\mathcal O}(d^{5n} log(s))\). Comparable bounds are obtained for the point location problem.

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Authors and Affiliations

  1. Dept. of Mathematics, University of Buenos Aires,  

    Rafael Grimson

  2. Dept. of Computer Science, Hasselt University,  

    Rafael Grimson

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  1. Rafael Grimson
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Editors and Affiliations

  1. GALAAD, Inria Méditerannée, 2004 route des Lucioles, 06902, Sophia Antipolis Cedex, France

    Bernard Mourrain

  2. Department of Computer Science, College Station, Texas A&M University, 77843-3112, TX, USA

    Scott Schaefer

  3. Institute of Computational Mathematics and Scientific and Engineering Computing, Chinese Academy of Science, 100080, Beijing, China

    Guoliang Xu

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Grimson, R. (2010). An Efficient Algorithm for the Sign Condition Problem in the Semi-algebraic Context. In: Mourrain, B., Schaefer, S., Xu, G. (eds) Advances in Geometric Modeling and Processing. GMP 2010. Lecture Notes in Computer Science, vol 6130. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13411-1_5

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  • DOI: https://doi.org/10.1007/978-3-642-13411-1_5

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-13410-4

  • Online ISBN: 978-3-642-13411-1

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