{"docId":86,"paperId":86,"url":"https:\/\/hrj.episciences.org\/86","doi":"10.46298\/hrj.2005.86","journalName":"Hardy-Ramanujan Journal","issn":"","eissn":"2804-7370","volume":[{"vid":39,"name":"Volume 28 - 2005"}],"section":[],"repositoryName":"Hal","repositoryIdentifier":"hal-00411308","repositoryVersion":1,"repositoryLink":"https:\/\/hal.archives-ouvertes.fr\/hal-00411308v1","dateSubmitted":"2015-03-03 16:13:33","dateAccepted":"2015-06-12 16:05:10","datePublished":"2005-01-01 08:00:00","titles":{"en":"Further Variations on the Six Exponentials Theorem."},"authors":["Waldschmidt, Michel"],"abstracts":{"en":"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."},"keywords":[{"en":"six exponentials theorem"},{"en":"rank of matrices with coefficients being linear forms in logarithm"},"11J81 (11J86 11J89)","[MATH.MATH-NT] Mathematics [math]\/Number Theory [math.NT]"]}