10.46298/hrj.2005.86
https://hrj.episciences.org/86
Waldschmidt, Michel
Michel
Waldschmidt
Further Variations on the Six Exponentials Theorem.
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.
episciences.org
six exponentials theorem
rank of matrices with coefficients being linear forms in logarithm
11J81 (11J86 11J89)
[MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT]
2015-06-12
2005-01-01
2005-01-01
en
journal article
https://hal.archives-ouvertes.fr/hal-00411308v1
2804-7370
https://hrj.episciences.org/86/pdf
VoR
application/pdf
Hardy-Ramanujan Journal
Volume 28 - 2005
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