Joint distribution of conjugate algebraic numbers: a random polynomial approach

Abstract

We count the algebraic numbers of fixed degree by their $\mathbf{w}$-weighted $l_p$-norm which generalizes the na\"ive height, the length, the Euclidean and the Bombieri norms. For non-negative integers $k,l$ such that $k+2l\leq n$ and a Borel subset $B\subset \mathbb{R}\times\mathbb{C}_+^l$ denote by $\Phi_{p,\mathbf{w},k,l}(Q,B)$ the number of ordered $(k+l)$-tuples in $B$ of conjugate algebraic numbers of degree $n$ and $\mathbf{w}$-weighted $l_p$-norm at most $ Q$. We show that $$ \lim_{ Q\to\infty}\frac{\Phi_{p,\mathbf{w},k,l}( Q,B)}{ Q^{n+1}}=\frac{{\mathrm{Vol}}_{n+1}(\mathbb{B}_{p,\mathbf{w}}^{n+1})}{2\zeta(n+1)}\int_B \rho_{p,\mathbf{w},k,l}(\mathbf{x},\mathbf{z}){\rm d}\mathbf{x}{\rm d}\mathbf{z}, $$ where ${\mathrm{Vol}}_{n+1}(\mathbb{B}_{p,\mathbf{w}}^{n+1})$ is the volume of the unit $\mathbf{w}$-weighted $l_p$-ball and $\rho_{p,\mathbf{w},k,l}$ will denote the correlation function of $k$ real and $l$ complex zeros of the random polynomial $\sum_{j=1}^n \frac{\eta_j}{w_j} z^j$, where $\eta_j $ are i.i.d. random variables with density $c_p e^{-|t|^p}$ for $0<p<\infty$ and with constant density on $[-1,1]$ for $p=\infty$. If the boundary of $B$ is of Lipschitz type, we also estimate the rate of convergence. We give an explicit formula for $\rho_{p,\mathbf{w},k,l}$, 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.