Simultaneous Diophantine approximation in two metrics and the distance between conjugate algebraic numbers in \(\mathbb C\times\mathbb Q_p\) (Q663562)

The authors provide a lower bound for the number of irreducible integer polynomials of degree at least three which have close conjugate roots in the complex and \(p\)-adic fields simultaneously. The central theorem of the paper -- Theorem 3 -- is a substantial improvement of a result of \textit{N. Budarina} et al. [Acta Math. Sinica (2012) in press]. Some of the methods used for proving it are taken from \textit{V. Beresnevich} et al. [Compositio Math. 146, 1165--1179 (2010; Zbl 1206.11091)] and from \textit{V. Bernik} et al. [Acta Arith. 133, 375--390 (2008; Zbl 1222.11096); Lith. Math. J. 48, No. 4, 380--396 (2008; Zbl 1228.11111)].