Authors: Blagojević, Pavle 
Dimitrijević-Blagojević, Aleksandra 
Ziegler, Günter
Title: Polynomial partitioning for several sets of varieties
Journal: Journal of Fixed Point Theory and Applications
Volume: 19
Issue: 3
First page: 1653
Last page: 1660
Issue Date: 1-Sep-2017
Rank: M21
ISSN: 1661-7738
DOI: 10.1007/s11784-016-0322-z
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≥ 1 and any collection of sets Γ 1, … , Γ j of low-degree k-dimensional varieties in Rn, there exists a non-zero polynomial p∈ R[ X1, … , Xn] of degree at most D, so that each connected component of Rn\ Z(p) intersects O(jDk-n| Γ i|) varieties of Γ i, simultaneously for every 1 ≤ i≤ 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.
Publisher: Springer Link
Project: Advanced Techniques of Cryptology, Image Processing and Computational Topology for Information Security 

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