Further Variations on the Six Exponentials Theorem.

Abstract

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.