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Journal of Fixed Point Theory and Applications

, Volume 19, Issue 3, pp 1653–1660 | Cite as

Polynomial partitioning for several sets of varieties

  • Pavle V. M. Blagojević
  • Aleksandra S. Dimitrijević Blagojević
  • Günter M. ZieglerEmail author
Article
  • 132 Downloads

Abstract

We give a new, systematic proof for a recent result of Larry Guth and thus also extend the result to a setting with several families of varieties: For any integer \(D\ge 1\) and any collection of sets \(\Gamma _1,\ldots ,\Gamma _j\) of low-degree k-dimensional varieties in \(\mathbb {R}^n\), there exists a non-zero polynomial \(p\in \mathbb {R}[X_1,\ldots ,X_n]\) of degree at most D, so that each connected component of \(\mathbb {R}^n{\setminus }Z(p)\) intersects \(O(jD^{k-n}|\Gamma _i|)\) varieties of \(\Gamma _i\), simultaneously for every \(1\le i\le j\). For \(j=1\), we recover the original result by Guth. Our proof, via an index calculation in equivariant cohomology, shows how the degrees of the polynomials used for partitioning are dictated by the topology, namely, by the Euler class being given in terms of a top Dickson polynomial.

Mathematics Subject Classification

55N25 52C45 

Notes

Acknowledgments

We are grateful to Josh Zahl and to the referee of JFPTA for very valuble remarks.

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Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Institute of MathematicsFU BerlinBerlinGermany
  2. 2.Mathematical Institute SASABeogradSerbia

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