The joint distribution of real conjugate algebraic numbers

In the talk, we will discuss the distribution of real algebraic numbers and correlations between conjugate algebraic numbers.
The degree $\deg(\alpha)$ and height $H(\alpha)$ of an algebraic number $\alpha$ are defined accordingly as the degree and height of its minimal polynomial, that is, a polynomial of the minimal degree with coprime integer coefficients that vanishes at $\alpha$. The height of a polynomial is the maximum of the absolute values of its coefficients.


For $B\subset\mathbb{R}^k$ denote by $\Phi_k(Q,B)$ the number of ordered $k$-tuples in $B$ of real conjugate algebraic numbers of degree $\leqslant n$ and naive height $\leqslant Q$. The asymptotics of $\Phi_1(Q,B)$ (as $Q\to\infty$) was earlier found in [1] for arbitrary fixed $n$. For $k\ge 2$, the following asymptotic formula was recently proved in [2]:
$$ \Phi_k(Q;B) = \frac{(2Q)^{n+1}}{2\zeta(n+1)} \int\limits_{B} \chi_k(\mathbf{x}) \prod_{1\le i < j \le k} |x_i - x_j|\,d\mathbf{x} + O\left(Q^n\right),\quad Q\to \infty, $$
where the function $\chi_k$ is continuous in $\mathbb{R}^k$ and can be given explicitly, $\zeta(\cdot)$ is the Riemann zeta function. If $n=2$, then the additional factor $\log Q$ appears in the reminder term.


The talk is based on the joint paper [2] by F. Götze, D. Zaporozhets and the speaker.


[1] D. Koleda, On the density function of the distribution of real algebraic numbers.Preprint, arXiv:1405.1627, 2014.


[2] F.Götze, D. Koleda, and D. Zaporozhets, Correlations between real conjugate algebraic numbers. Chebyshevskii Sb., 16:(4) (2015), p. 91–99.