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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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The set of zeroes of an “almost polynomial” function
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by Y. Yomdin PDF
Proc. Amer. Math. Soc. 90 (1984), 538-542 Request permission

Abstract:

Let $f$ be a smooth function on the unit $n$-dimensional ball, with the ${C^0}$-norm, equal to one. We prove that if for some $k \geqslant 2$, the norm of the $k$ th derivative of $f$ is bounded by ${2^{ - k - 1}}$, then the set of zeroes $Y$ of $f$ is similar to that of a polynomial of degree $k - 1$. In particular, $Y$ is contained in a countable union of smooth hypersurfaces; "many" straight lines cross $Y$ in not more than $k - 1$ points, and the $n - 1$-volume of $Y$ is bounded by a constant, depending only on $n$ and $k$.
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Additional Information
  • © Copyright 1984 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 90 (1984), 538-542
  • MSC: Primary 41A65; Secondary 41A10
  • DOI: https://doi.org/10.1090/S0002-9939-1984-0733402-5
  • MathSciNet review: 733402