{\L}ojasiewicz exponent and pluricomplex Green function on algebraic sets

We study pluricomplex Green functions on algebraic sets. Let $f$ be a proper holomorphic mapping between two algebraic sets. Given a compact set $K$ in the range of $f$, we show how to estimate the pluricomplex Green functions of $K$ and of $f^{-1}(K)$ in terms of each other, the {\L}ojasiewicz exponent of $f$ and the growth exponent of $f$. This result leads to explicit examples of pluricomplex Green functions on algebraic sets. We also present an enhanced version of the Bernstein-Walsh polynomial inequality specific to algebraic sets. This article provides a theoretical framework for future investigations of the rate of polynomial approximation of holomorphic functions on algebraic sets in the style of Bernstein-Walsh-Siciak theorem

Chebyshev admissible meshes and Lebesgue constants of complex polynomial projections

We construct admissible polynomial meshes on piecewise polynomial or trigonometric curves of the complex plane, by mapping univariate Chebyshev points. Such meshes can be used for polynomial least-squares, for the extraction of Fekete-like and Leja-like interpolation sets, and also for the evaluation of their Lebesgue constants

Polynomial inequalities on certain algebraic hypersurfaces

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