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 naïve 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 ${\Phi }_{p,\mathbf{w},k,l}(Q,B)$ 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 $$\underset{Q\to \infty }{\lim }\frac{{\Phi }_{p,\mathbf{w},k,l}(Q,B)}{Q^{n+1}}=\frac{{\mathrm{Vol}}_{n+1}({\mathbb{B}}_{p,\mathbf{w}}^{n+1})}{2\zeta (n+1)}{\int }_B{\rho }_{p,\mathbf{w},k,l}(\mathbf{x},\mathbf{z})\mathrm{d}\mathbf{x}\mathrm{d}\mathbf{z},$$ 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.

to be published in Adv. Math. arXiv admin note: text overlap with arXiv:1610.03610