QUANTUM CHAOTIC DYNAMICS AND RANDOM POLYNOMIALS

We investigate the distribution of roots of polynomials of high degree with random coefficients which, among others, appear naturally in the context of ''quantum chaotic dynamics.'' It is shown that under quite general conditions their roots tend to concentrate near the unit circ le in the complex plane. In order to further increase this tendency, w e study in detail the particular case of self-inversive random polynom ials and show that for them a finite portion of all roots lies exactly on the unit circle. Correlation functions of these roots are also com puted analytically, and compared to the correlations of eigenvalues of random matrices. The problem of ergodicity of chaotic wavefunctions i s also considered. For that purpose we introduce a family of random po lynomials whose roots spread uniformly over phase space. While these r esults are consistent with random matrix theory predictions, they prov ide a new and different insight into the problem of quantum ergodicity Special attention is devoted to the role of symmetries in the distrib ution of roots of random polynomials.