Sur quelques résultats récents de transcendance. (On some recent transcendence results). (French) Zbl 0762.11027

Journées arithmétiques, Exp. Congr., Luminy/Fr. 1989, Astérisque 198-200, 209-230 (1991).
[For the entire collection see Zbl 0743.00058.]
In the first part the author explains some recent results obtained by Gel’fond-Schneider-Baker’s method: lower bound of the rank of a matrix with logarithm coefficients [M. Waldschmidt, New advances in transcendence theory, Proc. Symp. Durham 1986, 375-398 (1988; Zbl 0659.10035)]; applications to the Leopoldt’s conjecture [M. Laurent, J. Reine Angew. Math. 399, 81-108 (1989; Zbl 0666.12001)] and to large transcendence degrees for families of exponentials ]G. Diaz, J. Number Theory 31, 1-23 (1989; Zbl 0661.10047)[.
The main tools refered to are: effective Nullstellensatz [works of W. D. Brownawell; and J. Kollar, J. Am. Math. Soc. 1, 963-975 (1988; Zbl 0682.14001)]; P. Philippon’s criterion for algebraic independence [Publ. Inst. Hautes Etud. Sci. 64, 5-52 (1988; Zbl 0615.10044)]; zero estimates in commutative algebraic groups [P. Philippon, Bull. Soc. Math. Fr. 114, 355-383 (1986; Zbl 0617.14001)]; concerning this last item, the author gives some recent results about a conjecture of S. Lang [M. Hindry, Invent. Math. 94, 575-603 (1988; Zbl 0638.14026)].
In the second part the author illustrates his new transcendence method by giving as example the proof of the so called six exponentials theorem: instead of solving the linear system with the auxiliary function as used in the classical method of Gel’fond-Schneider, and of applying a Siegel’s lemma, he studies directly the minors of the system (interpolation determinants).


11J81 Transcendence (general theory)
11J86 Linear forms in logarithms; Baker’s method
11R27 Units and factorization
11J89 Transcendence theory of elliptic and abelian functions
11E99 Forms and linear algebraic groups