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On the heights of totally \(p\)-adic numbers. (English. French summary) Zbl 1297.11073
Let \(K\) be a number field and let \(S\) be the set of places of \(K\). For each \(v \in S\) let \(L_v\) be the Galois closure of \(K_v\). The number \(\alpha\) is said to be totally in \(L_S/K\) if for each \(v \in S\) all the \(K\)-Galois conjugates of \(\alpha\) lie in \(L_v\). Generalizing an earlier result of E. Bombieri and U. Zannier [Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl. 12, No. 1, 5–14 (2001; Zbl 1072.11077)], the author proves the inequality \[ \liminf_{\alpha \in O_L} h(\alpha) \geq \frac{1}{2} \sum_{v \in S} N_v \frac{\log p_v}{e_v (q_v^{f_v}-1)}, \] where \(S\) is a set of nonarchimedean places of \(K\), \(L\) is the field of all totally \(L_S/K\) numbers, \(O_L\) is its ring of integers, \(N_v=[K_v:\mathbb{Q}_v]/[K:\mathbb{Q}]\), \(e_v\), \(f_v\) denote the ramification and inertial degrees of \(L_v/K_v\), respectively, \(q_v\) is the order of the residue field of \(K_v\), and \(p_v\) is a rational prime above which \(v\) lies. A corresponding upper bound (differing form the lower bound only by the absence of the factor \(1/2\)) is also given. The author conjectures that the upper bound is sharp.

11G50 Heights
11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure
37P30 Height functions; Green functions; invariant measures in arithmetic and non-Archimedean dynamical systems
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