Polynomial partitioning for several sets of varieties

@article{Blagojevic2016PolynomialPF,
  title={Polynomial partitioning for several sets of varieties},
  author={Pavle V. M. Blagojevic and Aleksandra Dimitrijevic Blagojevic and G{\"u}nter M. Ziegler},
  journal={Journal of Fixed Point Theory and Applications},
  year={2016},
  volume={19},
  pages={1653-1660}
}
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 $$D\ge 1$$D≥1 and any collection of sets $$\Gamma _1,\ldots ,\Gamma _j$$Γ1,…,Γj of low-degree k-dimensional varieties in $$\mathbb {R}^n$$Rn, there exists a non-zero polynomial $$p\in \mathbb {R}[X_1,\ldots ,X_n]$$p∈R[X1,…,Xn] of degree at most D, so that each connected component of $$\mathbb {R}^n{\setminus }Z(p)$$Rn\Z(p) intersects… 
Generalizations of Joints Problem
We generalize the joints problem to sets of varieties and prove almost sharp bound on the number of joints. As a special case, given a set of $N$ $2$-planes in $\mathbb{R}^6$, the number of points at
Borsuk--Ulam theorems for elementary abelian 2-groups
Let G be a compact Lie group and let U and V be finite-dimensional real G-modules with V G = 0. A theorem of Marzantowicz, de Mattos and dos Santos estimates the covering dimension of the zero-set of
The discrete yet ubiquitous theorems of Carathéodory, Helly, Sperner, Tucker, and Tverberg
We discuss five discrete results: the lemmas of Sperner and Tucker from combinatorial topology and the theorems of Carath\'eodory, Helly, and Tverberg from combinatorial geometry. We explore their
Joints of Varieties
We generalize the Guth–Katz joints theorem from lines to varieties. A special case says that N planes (2-flats) in 6 dimensions (over any field) have $$O(N^{3/2})$$ O ( N 3 / 2 ) joints, where a

References

SHOWING 1-10 OF 10 REFERENCES
Polynomial partitioning for a set of varieties
  • L. Guth
  • Mathematics
    Mathematical Proceedings of the Cambridge Philosophical Society
  • 2015
Abstract Given a set Γ of low-degree k-dimensional varieties in $\mathbb{R}$ n , we prove that for any D ⩾ 1, there is a non-zero polynomial P of degree at most D so that each component of
Generalizations of Joints Problem
We generalize the joints problem to sets of varieties and prove almost sharp bound on the number of joints. As a special case, given a set of $N$ $2$-planes in $\mathbb{R}^6$, the number of points at
On the Erdős distinct distances problem in the plane
In this paper, we prove that a set of N points in R 2 has at least c N log N distinct distances, thus obtaining the sharp exponent in a problem of Erd} os. We follow the setup of Elekes and Sharir
An ideal-valued cohomological index theory with applications to Borsuk—Ulam and Bourgin—Yang theorems
Abstract Numerical-valued cohomological index theories for G-pairs (X, A) over B, where G is a compact Lie group, have proved useful in critical point theory and in proving Borsuk—Ulam and
Cohomology of finite groups
The cohomology of groups has, since its beginnings in the 1920s and 1930s, been the stage for significant interaction between algebra and topology and has led to the creation of important new fields
An Incidence Theorem in Higher Dimensions
TLDR
The polynomial ham sandwich theorem is used to prove almost tight bounds on the number of incidences between points and k-dimensional varieties of bounded degree in Rd.
Using The Borsuk Ulam Theorem Lectures On Topological Methods In Combinatorics And Geometry
Elementary TopologyCohomology of SheavesCohomology OperationsA Basic Course in Algebraic TopologyAn Illustrated Introduction to Topology and HomotopyApplications of Algebraic TopologyJerusalem
Using The Borsuk-Ulam Theorem
Fadell and Sufian Y . Husseini , An ideal - valued cohomological index theory with applications to Borsuk – Ulam and Bourgin – Yang theorems , Ergodic Theory Dynam
  • Milgram , Cohomology of Finite Groups
  • 2004