Distribution of complex algebraic numbers (Q2832807)

In the paper under review, the authors give a formula for the number of algebraic numbers of height \(\leq Q\) and degree \(n\) in a bounded complex region \(\Omega\). It is assumed that \(\Omega\) does not intersect the real axis and that its boundary consists of a finite number of algebraic curves. Denoting the above number by \(\Psi_n(Q,\Omega)\), their formula is NEWLINE\[NEWLINE \Psi_n(Q,\Omega)=\frac{Q^{n+1}}{\zeta(n+1)} \int_{\Omega} \psi(z) \nu(dz)+O(Q^n), NEWLINE\]NEWLINE where \(\nu\) is the Lebesgue measure in the complex plane. The limit density \(\psi(z)\) is given a by a complicated function involving an integral over an \(n-1\) dimensional real region. The implicit constant in the big-O notation depends on \(n\), the number of algebraic curves which form the boundary of \(\Omega\) and their degrees. NEWLINENEWLINENEWLINEThe authors also give some estimates for the function \(\psi\) both near the real axis and away from it in the region \(|z|\geq 1\). The proof of the main result uses known bounds on the number of reducible polynomials of height \(\leq Q\) and degree \(n\), as well as known estimates for the number of integer points in the region \(tA\) as a function of the parameter \(t\) as \(t\to\infty\), where \(A\subset {\mathbb R}^d\) has boundary consisting of finitely many algebraic surfaces only.