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Simultaneous Diophantine approximation in two metrics and the distance between conjugate algebraic numbers in $$\mathbb C\times\mathbb Q_p$$. (English) Zbl 1239.11075
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 N. Budarina et al. [Acta Math. Sinica (2012) in press]. Some of the methods used for proving it are taken from V. Beresnevich et al. [Compositio Math. 146, 1165–1179 (2010; Zbl 1206.11091)] and from 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)].
##### MSC:
 11J13 Simultaneous homogeneous approximation, linear forms 11J83 Metric theory
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##### References:
 [1] Beresnevich, V.; Bernik, V.I.; Götze, F., The distribution of close conjugate algebraic numbers, Compositio math., 146, 1165-1179, (2010) · Zbl 1206.11091 [2] Bernik, V.; Götze, F.; Kukso, O., Lower bounds for the number of integral polynomials with given order of discriminants, Acta arith., 133, 375-390, (2008) · Zbl 1222.11096 [3] Bernik, V.; Götze, F.; Kukso, O., On the divisibility of the discriminant of an integral polynomial by prime powers, Lith. math. J., 48, 380-396, (2008) · Zbl 1228.11111 [4] N. Budarina, D. Dickinson, J. Yuan, On the number of polynomials with small discriminants in the Euclidean and $$p$$-adic metrics, Acta Math. Sinica (2012) (in press). · Zbl 1275.11109 [5] Y. Bugeaud, A. Dujella, Root separation for irreducible integer polynomials, Bull. Lond. Math. Soc. (2011) Advance Access, published October 2011 (doi:10.1112/blms/bdr085) in press. · Zbl 1273.11049 [6] Bugeaud, Y.; Mignotte, M., On the distance between roots of integer polynomials, Proc. edinb. math. soc. (2), 47, 553-556, (2004) · Zbl 1071.11016 [7] Evertse, J.H., Distances between conjugates of an algebraic number, Publ. math. debrecen, 65, 323-340, (2004) · Zbl 1073.11066 [8] Mahler, K., An inequality for the discriminant of a polynomial, Michigan math. J., 11, 257-262, (1964) · Zbl 0135.01702 [9] Sprindžuk, V., ()
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