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Proceedings of the American Mathematical Society

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

 
 

 

The set of zeroes of an “almost polynomial” function


Author: Y. Yomdin
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
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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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Article copyright: © Copyright 1984 American Mathematical Society