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Given an irreducible algebraic curves $A$ in ${ℂ}^N$, let $m_d$ be the dimension of the complex vector space of all holomorphic polynomials of degree at most $d$ restricted to $A$. Let $K$ be a nonpolar compact subset of $A$, and for each $d=1,2,...,$ choose $m_d$ points ${\left\{A_{dj}\right\}}_{j=1,...,m_d}$ in $K$. Finally, let ${\Lambda }_d$ be the $d$-th Lebesgue constant of the array $\left\{A_{dj}\right\}$; i.e., ${\Lambda }_d$ is the operator norm of the Lagrange interpolation operator $L_d$ acting on $C\left(K\right)$, where $L_d\left(f\right)$ is the Lagrange interpolating polynomial for $f$ of degree $d$ at the points ${\left\{A_{dj}\right\}}_{j=1,...,m_d}$. Using techniques of pluripotential...

For a regular, compact, polynomially convex circled set $K$ in ${\mathbf{C}}^2$, we construct a sequence of pairs $\{P_n,Q_n\}$ of homogeneous polynomials in two variables with $\mathrm{deg}{P}_n=$ $\mathrm{deg}{Q}_n=n$ such that the sets $K_n:=\{(z,w)\in {\mathbf{C}}^2:|P_n(z,w)|\le 1,|Q_n(z,w)|\le 1\}$ approximate $K$ and if $K$ is the closure of a strictly pseudoconvex domain the normalized counting measures associated to the finite set $\{P_n=Q_n=1\}$ converge to the pluripotential-theoretic Monge-Ampère measure for $K$. The key ingredient is an approximation theorem for subharmonic functions of logarithmic growth in one complex...