Characterization of compact subsets of algebraic varieties in terms of Bernstein type inequalities (Q2717545)

It is shown that in the class of compact sets \(K\) in \(\mathbb R^n\) with an analytic parameterization of order \(m\) (\(1\leq m\leq n\)), the sets with Zariski dimension \(m\) are exactly those which admit a Bernstein (or a van der Corput-Schaake) type inequality for tangential derivatives of (the traces of) polynomials on \(K\). In particular, by Hironaka's rectilinearization theorem, such a parameterization is admitted by compact subanalytic subsets of \(m\)-dimensional real-analytic submanifolds of \(\mathbb R^n\). Hence the above result covers a similar characterization of compact real-analytic manifolds given by \textit{L. Bos, N. Levenberg, P. Milman} and \textit{B. A. Taylor} [Indiana Univ. Math. J. 44, No. 1, 115-138 (1995; Zbl 0824.41015)].