Let $\Gamma$ be a non-pluripolar set in ${ℂ}^N$. Let $f$ be a function holomorphic in a connected open neighborhood $G$ of $\Gamma$. Let $\left\{P_n\right\}$ be a sequence of polynomials with $deg{P}_n\le d_n(d_n\<d_{n+1})$ such that$$\underset{n\to \infty }{lim\; sup}{|f\left(z\right)-P_n\left(z\right)|}^{1/d_n}\<1,z\in \Gamma .$$We show that if$$\underset{n\to \infty }{lim\; sup}{\left|P_n\left(z\right)\right|}^{1/d_n}\le 1,z\in E,$$where $E$ is a set in ${ℂ}^N$ such that the global extremal function $V_E\equiv 0$ in ${ℂ}^N$, then the maximal domain of existence $G_f$ of $f$ is one-sheeted, and$$\underset{n\to \infty }{lim\; sup}{\parallel f-P_n\parallel }_K^{\frac{1}{d_n}}\<1$$for every compact set $K\subset G_f$. If, moreover, the sequence $\{d_{n+1}/d_n\}$ is bounded then $G_f={ℂ}^N$.If $E$ is a closed set in ${ℂ}^N$ then $V_E\equiv 0$ if and only if each series of homogeneous polynomials ${\sum }_{j=0}^{\infty }{Q}_j$, for which some subsequence $\left\{s_{n_k}\right\}$...

The Siciak extremal function establishes an important link between polynomial approximation in several variables and pluripotential theory. This yields its numerous applications in complex and real analysis. Some of them can be found on a rich list drawn up by Klimek in his well-known monograph "Pluripotential Theory". The purpose of this paper is to supplement it by applications in constructive function theory.

The paper is concerned with the best constants in the Bernstein and Markov inequalities on a compact set $E\subset {ℂ}^N$. We give some basic properties of these constants and we prove that two extremal-like functions defined in terms of the Bernstein constants are plurisubharmonic and very close to the Siciak extremal function ${\Phi }_E$. Moreover, we show that one of these extremal-like functions is equal to ${\Phi }_E$ if E is a nonpluripolar set with $li{m}_{n\to \infty }Mₙ{\left(E\right)}^{1/n}=1$ where $Mₙ\left(E\right):=sup{\left|\right|\left|gradP\right|\left|\right|}_E/{\left|\right|P\left|\right|}_E$, the supremum is taken over all polynomials P of N variables of total...

We use our disc formula for the Siciak-Zahariuta extremal function to characterize the polynomial hull of a connected compact subset of complex affine space in terms of analytic discs.