Title: Joint distribution of conjugate algebraic numbers: a random polynomial approach,
Abstract: Given a polynomial $q(z):=a_0+a_1z+\dots+a_nz^n$ and a vector of positive
weights $\mathbf{w}=(w_0, w_1,\dots,w_n)$, define the $\mathbf{w}$-weighted
$l_p$-norm of $q$ as $$ l_{p,\mathbf{w}}[q]:=\left(\sum_{k=0}^{n}|w_k
a_k|^p\right)^{1/p},\quad p\in[1,\infty]. $$ Define the $\mathbf{w}$-weighted
$l_p$-norm of an algebraic number to be the $\mathbf{w}$-weighted $l_p$-norm of
its minimal polynomial. For non-negative integers $k,l$ such that $k+2l\leq n$
and a Borel subset $B\subset \mathbb{R}^k\times\mathbb{C}_+^l$ 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 $\rho_{p,\mathbf{w},k,l}$ shall
denote the correlation function of $k$ real and $l$ complex zeros of the random
polynomial $\sum_{k=1}^n \frac{\eta_k}{w_k} z^k$ for i.i.d. random variables
$\eta_k $ with density $c_p e^{|t|^p}$ for $p<\infty$ resp. with constant
density on $[-1,1]$ for $p=\infty$. We give an explicit formula for
$\rho_{p,\mathbf{w},k,l}$ which in the case $k+2l=n$ simplifies to $$
\rho_{p,\mathbf{w},n-2l,l}=\frac{2}{(n+1)\mathrm{Vol}_{n+1}(\mathbb{B}_{p,\mathbf{w}}^{n+1})}\,\frac{\sqrt{|\mathrm{D}[q]|}\phantom{1^n}}{(l_{p,\mathbf{w}}[q])^{n+1}},
$$ where $q$ is the monic polynomial whose zeros are the arguments of the
correlation function $\rho_{p,\mathbf{w},n-2l,l}$ and $\mathrm{D}[q]$ denotes
its discriminant.