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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
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

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