Joint distribution of conjugate algebraic numbers: A random polynomial approach

Abstract

We count the algebraic numbers of fixed degree by their w-weighted lp-norm which generalizes the naïve height, the length, the Euclidean and the Bombieri norms. For non-negative integers k,l such that k+2l≤n and a Borel subset B⊂R×C+l denote by Φp,w,k,l(Q,B) the number of ordered (k+l)-tuples in B of conjugate algebraic numbers of degree n and w-weighted lp-norm at most Q. We show thatlimQ→∞⁡Φp,w,k,l(Q,B)Qn+1=Voln+1(Bp,wn+1)2ζ(n+1)∫Bρp,w,k,l(x,z)dxdz, where Voln+1(Bp,wn+1) is the volume of the unit w-weighted lp-ball and ρp,w,k,l will denote the correlation function of k real and l complex zeros of the random polynomial ∑j=1nηjwjzj, where ηj are i.i.d. random variables with density cpe−|t|p for 0<p<∞ and with constant density on [−1,1] for p=∞. If the boundary of B is of Lipschitz type, we also estimate the rate of convergence. We give an explicit formula for ρp,w,k,l, which in the case k+2l=n has a very simple form. To this end, we obtain a general formula for the correlations between real and complex zeros of a random polynomial with arbitrary independent absolutely continuous coefficients.