3 Search Results for "de Zeeuw, Frank"


Document
Testing Polynomials for Vanishing on Cartesian Products of Planar Point Sets

Authors: Boris Aronov, Esther Ezra, and Micha Sharir

Published in: LIPIcs, Volume 164, 36th International Symposium on Computational Geometry (SoCG 2020)


Abstract
We present subquadratic algorithms, in the algebraic decision-tree model of computation, for detecting whether there exists a triple of points, belonging to three respective sets A, B, and C of points in the plane, that satisfy a certain polynomial equation or two equations. The best known instance of such a problem is testing for the existence of a collinear triple of points in A×B×C, a classical 3SUM-hard problem that has so far defied any attempt to obtain a subquadratic solution, whether in the (uniform) real RAM model, or in the algebraic decision-tree model. While we are still unable to solve this problem, in full generality, in subquadratic time, we obtain such a solution, in the algebraic decision-tree model, that uses only roughly O(n^(28/15)) constant-degree polynomial sign tests, for the special case where two of the sets lie on one-dimensional curves and the third is placed arbitrarily in the plane. Our technique is fairly general, and applies to any other problem where we seek a triple that satisfies a single polynomial equation, e.g., determining whether A× B× C contains a triple spanning a unit-area triangle. This result extends recent work by Barba et al. [Luis Barba et al., 2019] and by Chan [Timothy M. Chan, 2020], where all three sets A, B, and C are assumed to be one-dimensional. While there are common features in the high-level approaches, here and in [Luis Barba et al., 2019], the actual analysis in this work becomes more involved and requires new methods and techniques, involving polynomial partitions and other related tools. As a second application of our technique, we again have three n-point sets A, B, and C in the plane, and we want to determine whether there exists a triple (a,b,c) ∈ A×B×C that simultaneously satisfies two real polynomial equations. For example, this is the setup when testing for the existence of pairs of similar triangles spanned by the input points, in various contexts discussed later in the paper. We show that problems of this kind can be solved with roughly O(n^(24/13)) constant-degree polynomial sign tests. These problems can be extended to higher dimensions in various ways, and we present subquadratic solutions to some of these extensions, in the algebraic decision-tree model.

Cite as

Boris Aronov, Esther Ezra, and Micha Sharir. Testing Polynomials for Vanishing on Cartesian Products of Planar Point Sets. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 8:1-8:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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@InProceedings{aronov_et_al:LIPIcs.SoCG.2020.8,
  author =	{Aronov, Boris and Ezra, Esther and Sharir, Micha},
  title =	{{Testing Polynomials for Vanishing on Cartesian Products of Planar Point Sets}},
  booktitle =	{36th International Symposium on Computational Geometry (SoCG 2020)},
  pages =	{8:1--8:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-143-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{164},
  editor =	{Cabello, Sergio and Chen, Danny Z.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.8},
  URN =		{urn:nbn:de:0030-drops-121666},
  doi =		{10.4230/LIPIcs.SoCG.2020.8},
  annote =	{Keywords: Algebraic decision tree, Polynomial partition, Collinearity testing, 3SUM-hard problems, Polynomials vanishing on Cartesian products}
}
Document
Polynomials Vanishing on Cartesian Products: The Elekes-Szabó Theorem Revisited

Authors: Orit E. Raz, Micha Sharir, and Frank de Zeeuw

Published in: LIPIcs, Volume 34, 31st International Symposium on Computational Geometry (SoCG 2015)


Abstract
Let F in Complex[x,y,z] be a constant-degree polynomial, and let A,B,C be sets of complex numbers with |A|=|B|=|C|=n. We show that F vanishes on at most O(n^{11/6}) points of the Cartesian product A x B x C (where the constant of proportionality depends polynomially on the degree of F), unless F has a special group-related form. This improves a theorem of Elekes and Szabo [ES12], and generalizes a result of Raz, Sharir, and Solymosi [RSS14a]. The same statement holds over R. When A, B, C have different sizes, a similar statement holds, with a more involved bound replacing O(n^{11/6}). This result provides a unified tool for improving bounds in various Erdos-type problems in combinatorial geometry, and we discuss several applications of this kind.

Cite as

Orit E. Raz, Micha Sharir, and Frank de Zeeuw. Polynomials Vanishing on Cartesian Products: The Elekes-Szabó Theorem Revisited. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 522-536, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


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@InProceedings{raz_et_al:LIPIcs.SOCG.2015.522,
  author =	{Raz, Orit E. and Sharir, Micha and de Zeeuw, Frank},
  title =	{{Polynomials Vanishing on Cartesian Products: The Elekes-Szab\'{o} Theorem Revisited}},
  booktitle =	{31st International Symposium on Computational Geometry (SoCG 2015)},
  pages =	{522--536},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-83-5},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{34},
  editor =	{Arge, Lars and Pach, J\'{a}nos},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SOCG.2015.522},
  URN =		{urn:nbn:de:0030-drops-51031},
  doi =		{10.4230/LIPIcs.SOCG.2015.522},
  annote =	{Keywords: Combinatorial geometry, incidences, polynomials}
}
Document
Bisector Energy and Few Distinct Distances

Authors: Ben Lund, Adam Sheffer, and Frank de Zeeuw

Published in: LIPIcs, Volume 34, 31st International Symposium on Computational Geometry (SoCG 2015)


Abstract
We introduce the bisector energy of an n-point set P in the real plane, defined as the number of quadruples (a,b,c,d) from P such that a and b determine the same perpendicular bisector as c and d. If no line or circle contains M(n) points of P, then we prove that the bisector energy is O(M(n)^{2/5}n^{12/5} + M(n)n^2). We also prove the lower bound M(n)n^2, which matches our upper bound when M(n) is large. We use our upper bound on the bisector energy to obtain two rather different results: (i) If P determines O(n / sqrt(log n)) distinct distances, then for any 0 < a < 1/4, either there exists a line or circle that contains n^a points of P, or there exist n^{8/5 - 12a/5} distinct lines that contain sqrt(log n) points of P. This result provides new information on a conjecture of Erdös regarding the structure of point sets with few distinct distances. (ii) If no line or circle contains M(n) points of P, then the number of distinct perpendicular bisectors determined by P is min{M(n)^{-2/5}n^{8/5}, M(n)^{-1}n^2}). This appears to be the first higher-dimensional example in a framework for studying the expansion properties of polynomials and rational functions over the real numbers, initiated by Elekes and Ronyai.

Cite as

Ben Lund, Adam Sheffer, and Frank de Zeeuw. Bisector Energy and Few Distinct Distances. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 537-552, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


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@InProceedings{lund_et_al:LIPIcs.SOCG.2015.537,
  author =	{Lund, Ben and Sheffer, Adam and de Zeeuw, Frank},
  title =	{{Bisector Energy and Few Distinct Distances}},
  booktitle =	{31st International Symposium on Computational Geometry (SoCG 2015)},
  pages =	{537--552},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-83-5},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{34},
  editor =	{Arge, Lars and Pach, J\'{a}nos},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SOCG.2015.537},
  URN =		{urn:nbn:de:0030-drops-51086},
  doi =		{10.4230/LIPIcs.SOCG.2015.537},
  annote =	{Keywords: Combinatorial geometry, distinct distances, incidence geometry}
}
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