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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.

##### MSC:
 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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