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We give a new proof, relying on polynomial inequalities and some aspects of potential theory, of large deviation results for ensembles of random hermitian matrices.

For certain ensembles of random polynomials we give the expected value of the zero distribution (in one variable) and the expected value of the distribution of common zeros of m polynomials (in m variables).

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...

We prove several new results on the multivariate transfinite diameter and its connection with pluripotential theory: a formula for the transfinite diameter of a general product set, a comparison theorem and a new expression involving Robin's functions. We also study the transfinite diameter of the pre-image under certain proper polynomial mappings.

We show that a convex totally real compact set in ${ℂ}^n$ admits an extremal array for Kergin interpolation if and only if it is a totally real ellipse. (An array is said to be extremal for $K$ when the corresponding sequence of Kergin interpolation polynomials converges uniformly (on $K$) to the interpolated function as soon as it is holomorphic on a neighborhood of $K$.). Extremal arrays on these ellipses are characterized in terms of the distribution of the points and the rate of convergence is investigated....

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...