The conjecture due to Bertrand and Rodriguez Villegas asserts that the 1-norm of the nonzero element in an exterior power of the units of a number field has a certain lower bound. We prove this conjecture for the exterior square case when the number field is a real multi-quadratic Galois extension of any degree of the rationals.
La conjecture due à Bertrand et Rodriguez Villegas affirme que la norme 1 d’un élément non nul dans une puissance extérieure des unités d’un corps de nombres admet une certaine borne inférieure. Nous démontrons cette conjecture dans le cas du carré extérieur lorsque le corps de nombres est une extension galoisienne multiquadratique réelle, de degré quelconque, du corps des rationnels.
Révisé le :
Accepté le :
Publié le :
Keywords: Bertrand–Rodriguez Villegas conjecture, units
CC-BY-ND 4.0
Dohyeong Kim; Seungho Song. Bertrand’s and Rodriguez Villegas’ conjecture for real multi-quadratic Galois extensions of the rationals. Journal de théorie des nombres de Bordeaux, Tome 38 (2026) no. 1, pp. 57-69. doi: 10.5802/jtnb.1354
@article{JTNB_2026__38_1_57_0,
author = {Dohyeong Kim and Seungho Song},
title = {Bertrand{\textquoteright}s and {Rodriguez} {Villegas{\textquoteright}} conjecture for real multi-quadratic {Galois} extensions of the rationals},
journal = {Journal de th\'eorie des nombres de Bordeaux},
pages = {57--69},
year = {2026},
publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
volume = {38},
number = {1},
doi = {10.5802/jtnb.1354},
language = {en},
url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1354/}
}
TY - JOUR AU - Dohyeong Kim AU - Seungho Song TI - Bertrand’s and Rodriguez Villegas’ conjecture for real multi-quadratic Galois extensions of the rationals JO - Journal de théorie des nombres de Bordeaux PY - 2026 SP - 57 EP - 69 VL - 38 IS - 1 PB - Société Arithmétique de Bordeaux UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1354/ DO - 10.5802/jtnb.1354 LA - en ID - JTNB_2026__38_1_57_0 ER -
%0 Journal Article %A Dohyeong Kim %A Seungho Song %T Bertrand’s and Rodriguez Villegas’ conjecture for real multi-quadratic Galois extensions of the rationals %J Journal de théorie des nombres de Bordeaux %D 2026 %P 57-69 %V 38 %N 1 %I Société Arithmétique de Bordeaux %U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1354/ %R 10.5802/jtnb.1354 %G en %F JTNB_2026__38_1_57_0
[1] A bound for the exterior product of -units, Algebra Number Theory, Volume 18 (2024) no. 9, pp. 1589-1617 | Zbl | DOI | MR
[2] Le problème de Lehmer en dimension supérieure, J. Reine Angew. Math., Volume 513 (1999), pp. 145-179 | Zbl | DOI | MR
[3] Duality on tori and multiplicative dependence relations, J. Aust. Math. Soc., Volume 62 (1997) no. 2, pp. 198-216 | Zbl | DOI | MR
[4] Heights in Diophantine geometry, New Mathematical Monographs, 4, Cambridge University Press, 2006 | Zbl | MR
[5] On Bertrand’s and Rodriguez Villegas’ higher-dimensional Lehmer conjecture, Pac. J. Math., Volume 321 (2022) no. 1, pp. 119-165 (with an appendix by F. Rodriguez Villegas) | Zbl | DOI
[6] Ratios of regulators in totally real extensions of number fields, J. Number Theory, Volume 37 (1991) no. 3, pp. 288-297 | Zbl | DOI | MR
[7] Factorization of certain cyclotomic functions, Ann. Math., Volume 34 (1933) no. 3, pp. 461-479 | Zbl | DOI
[8] Around the unit circle—Mahler measure, integer matrices and roots of unity, Universitext, Springer, 2021, xx+438 pages | Zbl | DOI | MR
[9] Eine Regulatorabschätzung, Abh. Math. Semin. Univ. Hamb., Volume 47 (1978), pp. 95-106 | Zbl | DOI | MR
[10] On the product of the conjugates outside the unit circle of an algebraic number, Acta Arith., Volume 24 (1973), pp. 385-399 | Zbl | DOI | MR
[11] Ideale kleiner Norm in Idealklassen und eine Regulatorabschätzung, Invent. Math., Volume 62 (1981) no. 3, pp. 367-380 | Zbl | DOI | MR
Cité par Sources :