Random Polynomials Associated with Non-orthonormal Bases

This article addresses an equidistribution problem concerning the zeros of random polynomials on C^m, which are represented using a general basis that may not necessarily be orthonormal. We prove the equidistribution results which are more general than the previously acquired ones for non-discrete probability measures. More precisely, our result demonstrates that the equidistribution holds true even when the random coefficients in the basis representation are not independent and identically distributed (i. i. d. ), and moreover, they are not constrained to any particular probability distribution function. Main tools are borrowed from the capacity theory. In the context of random polynomial mappings from C^m to Cᵏ, we prove an equidistribution theorem for various codimensions 1 k m by applying the results from the capacity theory along with the codimension 1 analysis. Based on our current information, this result is general enough to encompass many previous ones in the literature regarding the asymptotic distribution of zeros of random polynomial mappings. We touch upon some well-known probability in this direction. Finally, by extending the concept of a sequence of asymptotically Bernstein-Markov measures to the setting of holomorphic line bundles over compact K\"ahler manifolds, we establish a global equidistribution theorem related to the zeros of systems of random holomorphic sections for large tensor powers of a fixed line bundle for any codimension k.