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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’skiń≠-type inequalities)
42B30 \(H^p\)-spaces
14H05 Algebraic functions and function fields in algebraic geometry
32U05 Plurisubharmonic functions and generalizations
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