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A Bernstein-type inequality for algebraic functions. (English) Zbl 0876.26015
Let $$V\subset\mathbb{R}^n$$ be an algebraic variety of pure dimension $$m$$ $$(1\leq m\leq n-1)$$. The purpose of this paper is to prove a local Bernstein inequality for certain families of algebraic functions that estimates the growth of an algebraic function bounded on a measurable subset of $$V$$ in a neighborhood of a regular point containing this subset. As a consequence, we prove that $$\log|p|_V|\in\text{BMO}(V)$$. Here $$p$$ is a real polynomial on $$\mathbb{R}^n$$ and $$V$$ is a compact algebraic manifold.

##### MSC:
 26C05 Real polynomials: analytic properties, etc. 41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities) 42B30 $$H^p$$-spaces 14H05 Algebraic functions and function fields in algebraic geometry 32U05 Plurisubharmonic functions and generalizations
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