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Geometric containment, common roots of polynomials and partial orders

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STACS 88 (STACS 1988)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 294))

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Abstract

The reduction to vector dominance of two unrelated problems is studied; the two problems are

  1. 1.

    Geometric containment: given a class of geometric figures, determine for any two figures in the class whether one is contained in the other (perhaps after translation, rotation and even reflection).

  2. 2.

    Common roots of polynomials: given a class of polynomials, determine for any two polynomials in the class whether they have positive common roots.

A general characterization of the relationship between geometric containment and properties of finite parametrizations of geometric figures is given in the form of an abstract theorem on partial orders. It is shown that the existing impossibility result for rectangles can be formulated as an instance of the abstract result.

The abstract result is then applied to some other classes of figures, and several instances are given using a known low-dimensional negative result to obtain a new higher-dimensional one.

The abstract theorem is then used to prove that domination of polynomials is not reducible to vector dominance even for the restricted class of quadratic polynomials.

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Author information

Authors and Affiliations

  1. School of Computer Science, Carleton University, Ottawa, Canada

    N. Santoro

  2. Faculty of Administration, University of Ottawa, Ottawa, Canada

    J. B. Sidney

  3. Department of Mathematics, University of Connecticut, Storrs, USA

    S. J. Sidney

  4. Depatment of Computer Science, University of Ottawa, Ottawa, Canada

    J. Urrutia

Authors
  1. N. Santoro
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  2. J. B. Sidney
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  3. S. J. Sidney
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  4. J. Urrutia
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Editor information

Robert Cori Martin Wirsing

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© 1988 Springer-Verlag Berlin Heidelberg

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Santoro, N., Sidney, J.B., Sidney, S.J., Urrutia, J. (1988). Geometric containment, common roots of polynomials and partial orders. In: Cori, R., Wirsing, M. (eds) STACS 88. STACS 1988. Lecture Notes in Computer Science, vol 294. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0035853

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  • DOI: https://doi.org/10.1007/BFb0035853

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-18834-6

  • Online ISBN: 978-3-540-48190-4

  • eBook Packages: Springer Book Archive

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