On the heights of totally p-adic numbers
Journal de Théorie des Nombres de Bordeaux, Tome 26 (2014) no. 1, pp. 103-109.

Bombieri et Zannier ont démontré des minorations et des majorations de la limite inférieure de la hauteur de Weil sur le corps des nombres totalement p-adiques et sur leurs généralisations. Dans notre étude nous utilisons des techniques de la théorie du potentiel pour généraliser les majorations de leur étude et, dans l’hypothèse d’intégralité, améliorer un peu plus les minorations.

Bombieri and Zannier established lower and upper bounds for the limit infimum of the Weil height in fields of totally p-adic numbers and generalizations thereof. In this paper, we use potential theoretic techniques to generalize the upper bounds from their paper and, under the assumption of integrality, to improve slightly upon their bounds.

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DOI : https://doi.org/10.5802/jtnb.861
Classification : 11G50,  11R06,  37P30
Mots clés : Weil height, totally p-adic, potential theory, Fekete-Szegő theorem.
@article{JTNB_2014__26_1_103_0,
     author = {Paul Fili},
     title = {On the heights of totally $p$-adic numbers},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {103--109},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {26},
     number = {1},
     year = {2014},
     doi = {10.5802/jtnb.861},
     zbl = {1297.11073},
     mrnumber = {3232769},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.861/}
}
Paul Fili. On the heights of totally $p$-adic numbers. Journal de Théorie des Nombres de Bordeaux, Tome 26 (2014) no. 1, pp. 103-109. doi : 10.5802/jtnb.861. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.861/

[1] M. Baker, A lower bound for average values of dynamical Green’s functions, Math. Res. Lett., 13 (2006),  245–257. | MR 2231115 | Zbl 1173.11041

[2] M. Baker and R. Rumely, Potential theory and dynamics on the Berkovich projective line, Mathematical Surveys and Monographs 159, American Mathematical Society, Providence, RI, (2010). | Zbl 1196.14002

[3] E. Bombieri and U. Zannier, A note on heights in certain infinite extensions of , Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 12 (2001),  5–14. | Zbl 1072.11077

[4] V. Flammang, Two new points in the spectrum of the absolute Mahler measure of totally positive algebraic integers, Math. Comp. 65 (1996),  307–311. | MR 1320894 | Zbl 0852.11058

[5] R. Rumely, The Fekete-Szegő theorem with splitting conditions. I, Acta Arith. 93 (2000),  99–116. | MR 1757183 | Zbl 0946.11025

[6] R. Rumely, The Fekete-Szegő theorem with splitting conditions. II, Acta Arith. 103 (2002),  347–410. | Zbl 1126.11342

[7] R. Rumely, Capacity theory on algebraic curves, Lecture Notes in Mathematics 1378, Springer-Verlag, Berlin, (1989). | MR 1009368 | Zbl 0679.14012

[8] C. J. Smyth, On the measure of totally real algebraic integers, J. Austral. Math. Soc. Ser. A 30 (1980/81),  137–149. | Zbl 0457.12001

[9] C. J. Smyth, On the measure of totally real algebraic integers. II, Math. Comp. 37 (1981),  205–208. | MR 616373 | Zbl 0475.12001