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Geometric problems solvable in single exponential time

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Applied Algebra, Algebraic Algorithms and Error-Correcting Codes (AAECC 1990)

Part of the book series: Lecture Notes in Computer Science ((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.

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Author information

Author notes

  1. Joos Heintz

    Present address: Working Group Noaï Fitchas. Instituto Argentino de Matematica conicet, Viamonte 1636, 1055, Buenos Aires, Argentina

Authors and Affiliations

  1. Working Group Noaï Fitchas. Instituto Argentino de Matematica conicet, Viamonte 1636, 1055, Buenos Aires, Argentina

    Teresa Krick & Pablo Solernó

  2. I.R.M.A.R. Université de Rennes I, 35042, Rennes Cedex, France

    Marie-Françoise Roy

Authors
  1. Joos Heintz
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  2. Teresa Krick
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  3. Marie-Françoise Roy
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  4. Pablo Solernó
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Editor information

Shojiro Sakata

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© 1991 Springer-Verlag Berlin Heidelberg

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Heintz, J., Krick, T., Roy, MF., Solernó, P. (1991). Geometric problems solvable in single exponential time. In: Sakata, S. (eds) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. AAECC 1990. Lecture Notes in Computer Science, vol 508. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-54195-0_35

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  • DOI: https://doi.org/10.1007/3-540-54195-0_35

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-54195-0

  • Online ISBN: 978-3-540-47489-0

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