# Strongest known version of Baker's theorem

The article I have checked for Baker's theorem is Waldschmidt's. But the article and the citations therein are from the time of '88. Question:

What is the the strongest known lower bound for Baker's theorem on linear forms on logarithms?

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The motivation(if it is required for answering) is for looking into Tijdemann's methods for Catalan conjecture. The p-adic part is added for curiosity. –  Anweshi Jul 25 '10 at 7:42
Request for longer tags: Here, for instance, transcendental-number-theory does not fit. –  Anweshi Jul 25 '10 at 7:48
That issue's already come up on meta; you can discuss it there if you like. For now, transcendental-nt works. –  Qiaochu Yuan Jul 25 '10 at 7:53
Ok, I implemented your suggestion. –  Anweshi Jul 25 '10 at 7:54
Wikipedia (article: Linear forms in logarithms) lists a version from 1993; no idea if that's been improved since then. –  Harry Altman Jul 25 '10 at 8:14

There is a big difference between linear forms in many logarithms and in two (or three) logarithms. The first case is covered in the archimedean case by the work of E. Matveev; Matveev's original works are hard even to specialists but there is a very nice survey [Yu. Nesterenko, Linear forms in logarithms of rational numbers, in Diophantine approximation (Cetraro, 2000), 53--106, Lecture Notes in Math., 1819, Springer, Berlin, 2003. MR2009829 (2004i:11082)]. The $p$-adic case was mostly done by Kunrui Yu. The best estimate for the case of two logarithms, which is of importance because of Tijdeman's application to Catalan's equation, is given in [M. Laurent, M. Mignotte et Y. Nesterenko, Formes linéaires en deux logarithmes et déterminants d'interpolation, J. Number Theory 55 (1995), no. 2, 285--321. MR1366574 (96h:11073)]. The latest news in the last direction (also in relation to Catalan's) are reviewed in [M. Mignotte, Linear forms in two and three logarithms and interpolation determinants. Diophantine equations, 151--166, Tata Inst. Fund. Res. Stud. Math., 20, Tata Inst. Fund. Res., Mumbai, 2008. MR1500224 (2010h:11119)]