Mathematische Zeitschrift

, Volume 277, Issue 3–4, pp 1105–1112 | Cite as

On sharp local turns of planar polynomials

Article
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Abstract

We show that for a real polynomial of degree \(n\) in two variables \(x\) and \(y\), any local “sharp turn” must have its “size” \({\gtrsim }e^{-Cn^{2}}\). We also show that there is indeed an example that has a sharp turn of size \({\lesssim }e^{-Cn}\). This gives a quite satisfactory answer to a problem raised by Guth. The formulation of the problem was inspired by applications of the polynomial method in the study of Kakeya conjecture.

Keywords

Real Polynomial Small Cube Polynomial Method Sharp Turn Vandermonde Determinant 
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Notes

Acknowledgments

The author was supported by mathematics department of Princeton University. He would like to thank Ben Yang for bringing this interesting problem to his attention, and Yuan Cao for helpful discussions which made him notice the advantage of considering the inner inscribed ball. He would also like to thank the helpful referee whose advice made the exposition of this paper improved.

References

  1. 1.
    Dvir, Z.: On the size of Kakeya sets in finite fields. J. Am. Math. Soc. 22, 1093–1097 (2009)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Guth, L.: The endpoint case of the Bennett–Carbery–Tao multilinear Kakeya conjecture. Acta Math. 205, 263–286 (2010)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Guth, L.: The Multilinear Kakeya Inequality, a file that can be downloaded from http://math.mit.edu/~lguth/PolyMethod/lect34

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of MathematicsPrinceton UniversityPrincetonUSA

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