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Geometric problems solvable in single exponential time

  • Joos Heintz
  • Teresa Krick
  • Marie-Françoise Roy
  • Pablo Solernó
Invited Contributions
Part of the Lecture Notes in Computer Science book series (LNCS, volume 508)

Abstract

Let S be a semialgebraic set given by a boolean combination of polynomial inequalities. We present an algorithmical method which solves in single exponential sequential time and polynomial parallel time, the following problems:
  • computation of the dimension of S.

  • computation of the number of semialgebraically connected components of S and construction of paths in S connecting points in the same component.

  • computation of the distance of S to another semialgebraic set and finding points realizing the distance if they exist.

  • computation of the “optical resolution” of S if S is closed (the pelotita and the bolón).

  • computation of integer Morse directions of S if S is a regular algebraic hypersurface.

The mentioned time bounds apply also to polynomial inequalities solving. As an application of our method we state an efficient Łojasiewicz inequality and an efficient finiteness theorem.

Keywords

Quantifier Elimination Polynomial Inequality Real Closed Field Semialgebraic Subset Quantifier Free Formula 
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 1991

Authors and Affiliations

  • Joos Heintz
    • 1
  • Teresa Krick
    • 1
  • Marie-Françoise Roy
    • 2
  • Pablo Solernó
    • 1
  1. 1.Working Group Noaï Fitchas. Instituto Argentino de Matematica conicetBuenos AiresArgentina
  2. 2.I.R.M.A.R. Université de Rennes IRennes CedexFrance

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