Discrete & Computational Geometry

, Volume 48, Issue 3, pp 499–517 | Cite as

Simple Proofs of Classical Theorems in Discrete Geometry via the Guth–Katz Polynomial Partitioning Technique

Article

Abstract

Recently Guth and Katz (arXiv:1011.4105, 2010) invented, as a step in their nearly complete solution of Erdős’s distinct distances problem, a new method for partitioning finite point sets in ℝd, based on the Stone–Tukey polynomial ham-sandwich theorem. We apply this method to obtain new and simple proofs of two well known results: the Szemerédi–Trotter theorem on incidences of points and lines, and the existence of spanning trees with low crossing numbers. Since we consider these proofs particularly suitable for teaching, we aim at self-contained, expository treatment. We also mention some generalizations and extensions, such as the Pach–Sharir bound on the number of incidences with algebraic curves of bounded degree.

Keywords

Incidences Algebraic techniques Crossing number Spanning tree with low crossing number Polynomial ham-sandwich Partitioning polynomial 

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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.School of Computer ScienceTel Aviv UniversityTel AvivIsrael
  2. 2.Department of Applied Mathematics and Institute of Theoretical Computer Science (ITI)Charles UniversityPraha 1Czech Republic
  3. 3.Institute of Theoretical Computer ScienceETH ZurichZurichSwitzerland
  4. 4.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA

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