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Abstract. For a regular compact set K in C m and a measure µ on K satisfying the Bernstein-Markov inequality, we consider the ensemble P N of polynomials of degree N , endowed with the Gaussian probability measure induced by L 2 (µ). We show that for large N , the simultaneous zeros of m polynomials in P N tend to concentrate around the Silov boundary of K; more precisely, their expected distribution is asymptotic to N m µ eq , where µ eq is the equilibrium measure of K. For the case where K is the unit ball, we give scaling asymptotics for the expected distribution of zeros as N → ∞.

There is a natural pluripotential-theoretic extremal function V K,Q associated to a closed subset K of C m and a realvalued, continuous function Q on K. We define random polynomials H n whose coefficients with respect to a related orthonormal basis are independent, identically distributed complex-valued random variables having a very general distribution (which includes both normalized complex and real Gaussian distributions) and we prove results on a.s. convergence of a sequence 1 n log |H n | pointwise and in L 1 loc (C m ) to V K,Q . In addition we obtain results on a.s. convergence of a sequence of normalized zero currents dd c 1 n log |H n | to dd c V K,Q as well as asymptotics of expectations of these currents. All these results extend to random polynomial mappings and to a more general setting of positive holomorphic line bundles over a compact Kähler manifold.

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