Multiplicity estimates on group varieties. (English) Zbl 0675.10024

Wüstholz, G.

Ann. Math. (2) 129, No. 3, 471-500 (1989).

The main result of this paper is a zero estimate on algebraic groups that takes into account multiplicities with respect to analytic subgroup. This estimate has been announced by the author some time ago (see for instance G. Wüstholz, Lect. Notes Math. 1068, 280-296 (1984; Zbl 0543.10024)]. We refer to D. W. Masser [Proc. Int. Congr. Math., Warszawa 1983, Vol. I, 493-502 (1984; Zbl 0564.10040)] for references to earlier works, especially by Brownawell, Masser, Nesterenko, as well as Masser and Wüstholz.
Compared with the previous joint works by Masser and Wüstholz, the new element of the proof is an estimate for the length of a primary ideal in terms of the order with respect to an analytic subgroup. More recent results are explained in D. Bertrand [Sémin. Bourbaki 1985/86, Exp. 652, Astérisque 145/146, 21-44 (1987; Zbl 0613.14001)]. In particular an improvement of the zero estimate, involving degrees of algebraic subgroups, is due to P. Philippon [Bull. Soc. Math. Fr. 114, 355-383 (1986; Zbl 0617.14001) and 115, 397-398 (1987; Zbl 0634.14001)].

Reviewer: M.Waldschmidt

MSC:

11J81	Transcendence (general theory)
14A05	Relevant commutative algebra
14L10	Group varieties

Keywords:

Nullstellensatz; zero estimate on algebraic groups; multiplicities; analytic subgroup

Citations:

Zbl 0543.10024; Zbl 0564.10040; Zbl 0613.14001; Zbl 0617.14001; Zbl 0634.14001