Correlations between real conjugate algebraic numbers

Abstract

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 $\leq n$ and naive height $\leq Q$. We show that $$ \Phi_k(Q;B) = \frac{(2Q)^{n+1}}{2\zeta(n+1)} \int_{B} \rho_k(\mathbf{x})\,d\mathbf{x} + O\left(Q^n\right),\quad Q\to \infty, $$ where the function $\rho_k$ will be given explicitly. If $n=2$, then an additional factor $\log Q$ appears in the reminder term.