Polynomial inequalities on algebraic sets (Q2717544)

An estimate of Siciak's extremal function for compact subsets of algebraic varieties in \(\mathbb C^n\) (resp. \(\mathbb R^n\)) is given. As an application, Bernstein-Walsh or (tangential) Markov type inequalities are established on subsets of an algebraic set in \(\mathbb C^n\) (resp. \(\mathbb R^n\)) that are images under non-degenerate analytic maps \(f\) of ``good'' compact subsets \(E\) of \(\mathbb C^k\). In particular, the authors show that if the Siciak function \(\Phi_E\) is Hölder continuous on \(E\) then there exists a constant \(C>0\) such that for any polynomial \(Q\in\mathbb C[z_1,\dots,z_n]\) of degree \(d\), one has NEWLINE\[NEWLINED_{T(t,v)}Q(z)|\leq Cd^r\|Q\|_{f(E)}NEWLINE\]NEWLINE where \(z=f(t)\) with \(t\in E\), \(1/r\) is the Hölder exponent of \(f\) and \(T(t,v)=D_vf(t)\), the derivative at \(t\) of the map \(f\) in direction \(v\in\mathbb S^{k-1}\).