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 -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 -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.
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},
mrnumber = {3232769},
zbl = {1297.11073},
language = {en},
url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.861/}
}
TY - JOUR AU - Paul Fili TI - On the heights of totally $p$-adic numbers JO - Journal de théorie des nombres de Bordeaux PY - 2014 SP - 103 EP - 109 VL - 26 IS - 1 PB - Société Arithmétique de Bordeaux UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.861/ DO - 10.5802/jtnb.861 LA - en ID - JTNB_2014__26_1_103_0 ER -
%0 Journal Article %A Paul Fili %T On the heights of totally $p$-adic numbers %J Journal de théorie des nombres de Bordeaux %D 2014 %P 103-109 %V 26 %N 1 %I Société Arithmétique de Bordeaux %U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.861/ %R 10.5802/jtnb.861 %G en %F JTNB_2014__26_1_103_0
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 | Zbl
[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
[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
[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 | Zbl
[5] R. Rumely, The Fekete-Szegő theorem with splitting conditions. I, Acta Arith. 93 (2000), 99–116. | MR | Zbl
[6] R. Rumely, The Fekete-Szegő theorem with splitting conditions. II, Acta Arith. 103 (2002), 347–410. | Zbl
[7] R. Rumely, Capacity theory on algebraic curves, Lecture Notes in Mathematics 1378, Springer-Verlag, Berlin, (1989). | MR | Zbl
[8] C. J. Smyth, On the measure of totally real algebraic integers, J. Austral. Math. Soc. Ser. A 30 (1980/81), 137–149. | Zbl
[9] C. J. Smyth, On the measure of totally real algebraic integers. II, Math. Comp. 37 (1981), 205–208. | MR | Zbl
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