Joint distribution of conjugate algebraic numbers: a random polynomial approach

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 $k+2l\le n$ and a Borel subset $B\subset \mathbb{R}\times {\mathbb{C}}_{+}^l$ denote by ${\mathrm{\Phi }}_{p,\mathbf{w},k,l}(Q,B)$ 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 $$\underset{Q\to \mathrm{\infty }}{\lim }\frac{{\mathrm{\Phi }}_{p,\mathbf{w},k,l}(Q,B)}{Q^{n+1}}=\frac{{\mathrm{V}\mathrm{o}\mathrm{l}}_{n+1}({\mathbb{B}}_{p,\mathbf{w}}^{n+1})}{2\zeta (n+1)}{\int }_B{\rho }_{p,\mathbf{w},k,l}(\mathbf{x},\mathbf{z})\mathrm{d}\mathbf{x}\mathrm{d}\mathbf{z},$$ where ${\mathrm{V}\mathrm{o}\mathrm{l}}_{n+1}({\mathbb{B}}_{p,\mathbf{w}}^{n+1})$ 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 $arei\mathrm{.}i\mathrm{.}d\mathrm{.}randomvariableswithdensity$c_p e^{-|t|^p}$for$0<p<\infty$andwithconstantdensityon$[-1,1]$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.