Correlations between real conjugate algebraic numbers

F. Götze ; D. Kaliada ; D. N. Zaporozhets

Čebyševskij sbornik, Tome 16 (2015) no. 4, pp. 90-99

Résumé

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 $$ \Phi_k(Q;B) = \frac{(2Q)^{n+1}}{2\zeta(n+1)} \int\limits_{B} \chi_k(\mathbf{x}) \prod_{1\le i j \le k} |x_i - x_j| d\mathbf{x} + O\left(Q^n\right),\quad Q\to \infty, $$ where the function $\chi_k$ is continuous in $\mathbb{R}^k$ and will be given explicitly. If $n=2$, then an additional factor $\log Q$ appears in the reminder term. This relation may be regarded as a "repulsion" of real algebraic conjugates from each other. The function $$ \rho_k(\mathbf{x}):= \chi_k(\mathbf{x}) \prod_{1\le i j \le k} |x_i - x_j| $$ coincides with a $k$-point correlation function of real zeros of a random polynomial of degree $n$ with independent coefficients uniformly distributed on $[-1,1]$. Bibliography: 18 titles.

Détail

Keywords: conjugate algebraic numbers, correlations between algebraic numbers, distribution of algebraic numbers, integral polynomial, random polynomial.