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

  • Rafael Grimson
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6130)

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.

Keywords

Arithmetic Operation Query Point Query Time Dialytic Method Arithmetic Circuit 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Rafael Grimson
    • 1
    • 2
  1. 1.Dept. of MathematicsUniversity of Buenos Aires 
  2. 2.Dept. of Computer ScienceHasselt University 

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