Random polynomials and their zeros

We investigate the distribution of zeros of random polynomials with independent and identically distributed standard normal coefficients in the complex domain, obtain explicit formulas for the density and mean distribution of the zeros and level-crossings, and inquire into the consequences of their asymptotical evaluations for a variety of orthogonal polynomials. In addition, we bridge a small gap in the method of proof devised by Shepp and Vanderbei. Our approach makes use of the Jacobians of functions of several complex variables and the mean ratio of complex normal random variables.