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Geometric containment, common roots of polynomials and partial orders

  • N. Santoro
  • J. B. Sidney
  • S. J. Sidney
  • J. Urrutia
Contributed Papers Geometrical Algorithms
  • 120 Downloads
Part of the Lecture Notes in Computer Science book series (LNCS, volume 294)

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.

Keywords

Partial Order Quadratic Polynomial Geometric Figure Isosceles Triangle Abstract Theorem 
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 1988

Authors and Affiliations

  • N. Santoro
    • 1
  • J. B. Sidney
    • 2
  • S. J. Sidney
    • 3
  • J. Urrutia
    • 4
  1. 1.School of Computer ScienceCarleton UniversityOttawaCanada
  2. 2.Faculty of AdministrationUniversity of OttawaOttawaCanada
  3. 3.Department of MathematicsUniversity of ConnecticutStorrsUSA
  4. 4.Depatment of Computer ScienceUniversity of OttawaOttawaCanada

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