DISTRIBUTION OF ROOTS OF RANDOM POLYNOMIALS

We consider polynomials of high degree with random coefficients which appear in the context of "quantum chaotic" dynamics and investigate various conditions under which their roots tend to concentrate near the unit circle in the complex plane. Correlation functions of roots are computed analytically. We also investigate a certain class of random polynomials whose roots cover, in a uniform way, the Riemann sphere. Special emphasis is devoted to the influence of symmetries.