# Geometric containment, common roots of polynomials and partial orders

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## Abstract

*vector dominance*of two unrelated problems is studied; the two problems are

- 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.
*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## Preview

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## References

- [BFM]B.S. Baker, S.J. Fortune, S.R. Mahaney, “Polygon containment under translation”,
*J. of Algorithms*7 (1986), 532–548.Google Scholar - [C1]B.M. Chazelle, "Optimal algorithms for computing depths and layers",
*Proc. 21th Allerton Conf. on Communication, Control and Computing*, Monticello, Oct. 1983, 427–436.Google Scholar - [C2]B.M. Chazelle, "The polygon containment problem",
*Advances of Computing Research 1*, 1–33.Google Scholar - [CK]D.A. Chand, S.S. Kapur, "An algorithm for convex polytopes",
*J. of the ACM 17*(1970), 78–86.Google Scholar - [EO]H. Edelsbrunner, M.H. Overmars, "On the equivalence of some rectangle problems",
*Information Processing Letters 14*(1982), 124–127.Google Scholar - [GY]R.L. Graham, F.F. Yao, "Finding the convex hull of a simple polygon",
*J. of Algorithms 4*(1983), 324–331.Google Scholar - [GNO]R.H. Guting, O. Nurmi, T. Ottman, "The direct dominance problem",
*Proc.1st Symp. on Computational Geometry*, June 1985, 81–88.Google Scholar - [KLP]H.T. Kung, F. Luccio, F.P. Preparata, "On finding the maxima of a set of vectors",
*J. of the ACM 22*(1975), 469–476.Google Scholar - [KS1]D.G. Kirkpatrick, R. Seidel, "The ultimate planar convex hull algorithm?",
*SIAM J. on Computing 15*(1986), 287–299.Google Scholar - [KS2]D.G. Kirkpatrick, R. Seidel, "Output-seize sensitive algorithms for finding maximal vectors",
*Proc.1st Symp. on Comput. Geometry*, June 1985, 89–96.Google Scholar - [LP]D.T. Lee, F.P. Preparata, "An improved algorithm for the rectangle enclosure problem",
*J. of Algorithms 3*(1982), 218–224.Google Scholar - [PH]F.P. Preparata, S.J. Hong, "Convex hulls of finite sets of points in two and three dimensions",
*Communications of the ACM*(1977), 87–93.Google Scholar - [Se]R. Seidel, "Constructing higher-dimensional convex hulls at logarithmic cost per face",
*Proc. 18th ACM Symp. on Theory of Computing*, May 1986, 404–413.Google Scholar - [Sh]N.K. Shinja,
*Control Systems*, Holt Reinehart & Wiston, 1986.Google Scholar - [SSSU]N. Santoro, J.B. Sidney, S.J. Sidney, J. Urrutia, "Geometric containment and vector dominance",
*Theoretical Computer Science*, to appear; preliminary version in*Proc.2nd Symp. on Theoretical Aspects of Computer Science*, Saarbrucken, Jan. 1985, 322–327.Google Scholar - [U]J. Urrutia, "Parial orders and Euclidian geometry", in
*Algorithms and Order*, (I. Rival ed.), Reidel, 1987.Google Scholar - [VW]V. Vaishnavi, D. Wood, "Data structures for rectangle containment and enclosure problems',
*Computer Graphics and Image processing 13*(1980), 372–384.Google Scholar