Correlations between real conjugate algebraic numbers

For $B\subset {\mathbb{R}}^k$ denote by ${\mathrm{\Phi }}_k(Q;B)$ the number of ordered $k$-tuples in $B$ of real conjugate algebraic numbers of degree $\le n$ and naive height $\le Q$. We show that $${\mathrm{\Phi }}_k(Q;B)=\frac{(2Q{)}^{n+1}}{2\zeta (n+1)}{\int }_B{\rho }_k(\mathbf{x})d\mathbf{x}+O\left(Q^n\right),Q\to \mathrm{\infty },$$ where the function ${\rho }_k$ will be given explicitly. If $n=2$, then an additional factor $\log Q$ appears in the reminder term.