episciences.org_86_1653159422
1653159422
episciences.org
raphael.tournoy+crossrefapi@ccsd.cnrs.fr
episciences.org
HardyRamanujan Journal
28047370
10.46298/journals/hrj
https://hrj.episciences.org
01
01
2005
Volume 28  2005
Further Variations on the Six Exponentials Theorem.
Michel
Waldschmidt
According to the Six Exponentials Theorem, a $2\times 3$ matrix whose entries $\lambda_{ij}$ ($i=1,2$, $j=1,2,3$) are logarithms of algebraic numbers has rank $2$, as soon as the two rows as well as the three columns are linearly independent over the field $\BbbQ$ of rational numbers. The main result of the present note is that one at least of the three $2\times 2$ determinants, viz. $$ \lambda_{21}\lambda_{12}\lambda_{11}\lambda_{22}, \quad \lambda_{22}\lambda_{13}\lambda_{12}\lambda_{23}, \quad \lambda_{23}\lambda_{11}\lambda_{13}\lambda_{21} $$ is transcendental.
01
01
2005
86
https://hal.archivesouvertes.fr/hal00411308v1
10.46298/hrj.2005.86
https://hrj.episciences.org/86

https://hrj.episciences.org/86/pdf