Hostname: page-component-8448b6f56d-gtxcr Total loading time: 0 Render date: 2024-04-24T12:34:11.294Z Has data issue: false hasContentIssue false
  • English
  • Français

Polynomial partitioning for a set of varieties

Published online by Cambridge University Press:  30 September 2015

LARRY GUTH
LARRY GUTH*
Affiliation:
MIT, Department of Mathematics, Building E18, Room 369, 77 Massachusetts Avenue, Cambridge, MA 02139-4307, U.S.A. e-mail: larry.guth.work@gmail.com
Get access Rights & Permissions [Opens in a new window]

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 $\mathbb{R}$n\Z(P) intersects O(Dk−n|Γ|) varieties of Γ.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 159 , Issue 3 , November 2015 , pp. 459 - 469
Copyright
Copyright © Cambridge Philosophical Society 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[BB] Barone, S. and Basu, S. Refined bounds on the number of connected components of sign conditions on a variety. Discrete Comput. Geom. Doi: 10.1007/s00454-011-9391-3 also in: arXiv:1104.0636v3 [math.CO]Google Scholar
[GK] Guth, L. and Katz, N. On the Erdős distinct distance problem in the plane. accepted for publication in Ann. of Math. arXiv:1011.4105Google Scholar
[GP] Guillemin, V. and Pollack, A. Differential Topology Reprint of the 1974 original (AMS Chelsea Publishing, Providence, RI, 2010) xviii+224 pp.Google Scholar
[KMS] Kaplan, H., Matousek, J. and Sharir, M. Simple proofs of classical theorems in discrete geometry via the Guth–Katz polynomial partitioning technique. Discrete Comput. Geom. 48 (2012), 499–517.Google Scholar
[KMSS] Kaplan, H., Matousek, J., Safernová, Z. and Sharir, M. Unit distances in three dimensions. Combinat. Probab. Comput. 21 (2012), 597–610. Also in arXiv:1107.1077.Google Scholar
[M] Milnor, J. On the Betti numbers of real varieties. Proc. Anel. Math. Soc. 15 (1964), 275–280.Google Scholar
[M2] Milnor, J. Topology from the Differentiable Viewpoint (Princeton University Press, Princeton, New Jersey, 1965.)Google Scholar
[SS] Sharir, M. and Solomon, N. Incidences between points and lines in four dimensions. Proc. 30th ACM Symp. on Computational Geometry (2014), to appear.Google Scholar
[SSZ] Sharir, M., Sheffer, A. and Zahl, J. Improved bounds for incidences between points and circles. Combinat. Probab. Comput. submitted. Also in Proc. 29th ACM Symp. on Computational Geometry (2013), 97–106. Also in arXiv:1208.0053.Google Scholar
[ST] Solymosi, J. and Tao, T. An incidence theorem in higher dimensions. Discrete Comput. Geom. 48 (2012), 255–280.Google Scholar
[StTu] Stone, A. and Tukey, J. Generalised sandwich theorems. Duke Math. J. 9 (1942), 356–359.Google Scholar
[T] Thom, R. Sur l'homologie des variétés algébriques réelles. Differential and Combinatorial Topology (Symposium in Honor of Marston Morse), Ed. Cairns, S.S. (Princeton University Press, 1965). 255–265.Google Scholar
[Z] Zahl, J. An improved bound on the number of point-surface incidences in three dimensions. Contrib. Discrete Math. 8 (2013), 100–121. Also in arXiv:1104.4987.Google Scholar