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 be a smooth function on the unit -dimensional ball, with the -norm, equal to one. We prove that if for some , the norm of the th derivative of is bounded by , then the set of zeroes of is similar to that of a polynomial of degree . In particular, is contained in a countable union of smooth hypersurfaces; "many" straight lines cross in not more than points, and the -volume of is bounded by a constant, depending only on and .

**[1]**M. Coste,*Ensembles semi-algebriques*, Geometrie Algebrique Reelle et Formes Quadratiques, Lecture Notes in Math., Springer, Berlin and New York, vol. 952, 1982, pp. 109-138. MR**683131 (84e:14023)****[2]**O. D. Kellogg,*On bounded polynomials in several variables*, Math. Z.**27**(1928), 55-64. MR**1544896****[3]**B. Malgrange,*Ideals of differentiable functions*, Oxford Univ. Press, London, 1966. MR**0212575 (35:3446)****[4]**H. Whitney,*Analytic extensions of differentiable functions defined in closed sets*, Trans. Amer. Math. Soc.**36**(1934), 63-89. MR**1501735****[5]**Y. Yomdin,*The geometry of critical and near-critical values of differentiable mappings*, Math. Ann.**264**(1983), 495-515. MR**716263 (86f:58017)**

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DOI:
https://doi.org/10.1090/S0002-9939-1984-0733402-5

Article copyright:
© Copyright 1984
American Mathematical Society