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3 - Polynomial Partitioning

Published online by Cambridge University Press:  17 March 2022

Adam Sheffer
Affiliation:
Bernard M. Baruch College, City University of New York
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Summary

In this chapter, we study our first new polynomial technique: polynomial partitioning. We first see the polynomial partitioning theorem. We use this theorem to derive an incidence bound between points and curves in the real plane. This bound generalizes the Szemerédi–Trotter theorem and the current best bound for the unit distances problem. In the second part of the chapter, we prove the polynomial partitioning theorem by using the ham sandwich theorem and Veronese maps. Finally, we use the point-curve incidence bound to obtain an upper bound for the number of lattice points that a curve can contain.

During the chapter we learn other important concepts, such as Warren’s theorem, incidence graphs, and various tricks for working with curves.

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Publisher: Cambridge University Press
Print publication year: 2022

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  • Polynomial Partitioning
  • Adam Sheffer, Bernard M. Baruch College, City University of New York
  • Book: Polynomial Methods and Incidence Theory
  • Online publication: 17 March 2022
  • Chapter DOI: https://doi.org/10.1017/9781108959988.004
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  • Polynomial Partitioning
  • Adam Sheffer, Bernard M. Baruch College, City University of New York
  • Book: Polynomial Methods and Incidence Theory
  • Online publication: 17 March 2022
  • Chapter DOI: https://doi.org/10.1017/9781108959988.004
Available formats
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Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • Polynomial Partitioning
  • Adam Sheffer, Bernard M. Baruch College, City University of New York
  • Book: Polynomial Methods and Incidence Theory
  • Online publication: 17 March 2022
  • Chapter DOI: https://doi.org/10.1017/9781108959988.004
Available formats
×