Archiv der Mathematik

, Volume 86, Issue 6, pp 561–568 | Cite as

Zero sets of polynomials in several variables

Original Paper


Let \(k, n \in \mathbb{N}\), where n is odd. We show that there is an integer N  =  N(k, n) such that for every n-homogeneous polynomial \(P : \mathbb{R}^N \rightarrow \mathbb{R}\) there exists a linear subspace \(X \hookrightarrow \mathbb{R}^N, \dim X = k\), such that P| X ≡ 0. This quantitative estimate improves on previous work of Birch et al., who studied this problem from an algebraic viewpoint. The topological method of proof presented here also allows us to obtain a partial solution to the Gromov-Milman problem (in dimension two) on an isometric version of a theorem of Dvoretzky.

Mathematics Subject Classification (2000).



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Copyright information

© Birkhäuser Verlag, Basel 2006

Authors and Affiliations

  1. 1.Department of Mathematical SciencesKent State UniversityKentUSA
  2. 2.Mathematical Institute of the Czech Academy of ScienceCZ-PrahaCzech Republic

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