×

zbMATH — the first resource for mathematics

A corollary to a theorem of Laurent-Mignotte-Nesterenko. (English) Zbl 0919.11051
Let \(b_{1}\) and \(b_{2}\) be two positive integers and \(\alpha_{1}\), \(\alpha_{2}\) two multiplicatively independent algebraic numbers. For \(i=1\) and \(i=2\), let \(\log\alpha_{i}\) be a value of the logarithm of \(\alpha_{i}\). Assume that the number \(\Lambda= b_{2}\log\alpha_{2}-b_{1}\log\alpha_{1}\) does not vanish. In a previous paper, M. Laurent, M. Mignotte and Yu. V. Nesterenko provided a sharp explicit lower bound for \(| \Lambda| \) [J. Number Theory 55, No. 2, 285-321 (1995; Zbl 0843.11036)]. The main result of this joint paper involved several auxiliary parameters (related to the construction of an interpolation determinant in the transcendence method), and a few more concrete consequences were given. These estimates have been used in several works of different authors who explicitly solved diophantine equations. For such applications it is of the utmost importance to get sharp numerical constants. In the present text under review, the author deduces from the main result of his previous joint paper a corollary which yields sharper lower bounds than the special cases already derived. The main point is an improved choice of the auxiliary parameters.

MSC:
11J86 Linear forms in logarithms; Baker’s method
PDF BibTeX XML Cite
Full Text: DOI EuDML