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An extended lemma of Dobrowolski and Smyth’s congruence. (English) Zbl 1265.11050

For a polynomial \(f(X) =\prod^d_{j=1}(X-a_j)\in {\mathbb Z}[X]\) and positive integer \(N\) denote by \(f_N(X)\) the polynomial \(\prod^d_{j=1}(X- a^N_j)\). It was shown by K. G. Hare, D. McKinnon and C. D. Sinclair [J. Théor. Nombres Bordx. 21, No. 1, 215–234 (2009; Zbl 1230.11035)] that for any integers \(m > n > 0\) and prime \(p\) the resultant of the polynomials \(f_{p^m}\) and \(f_{p^n}\) is divisible by \(p^{d(m+1)}\). The author gives a short proof of this result, utilizing a congruence proved by C. J. Smyth [Am. Math. Mon. 93, 469–471 (1986; Zbl 0602.10006)], which generalizes a lemma of E. Dobrowolski [Acta Arith. 34, 391–401 (1979; Zbl 0416.12001)]. As an application the lower bound \(M(f) > 1 + \exp(-c\sqrt{\log d})\) for the Mahler’s measure of \(f\) is deduced.

MSC:

11C08 Polynomials in number theory
11R04 Algebraic numbers; rings of algebraic integers
11R09 Polynomials (irreducibility, etc.)
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