Algebraic groups and small transcendence degree. I
This paper contains many results on algebraic independence of transcendental numbers of the following form: in certain specified sets of numbers (connected with values of exponential or elliptic functions, for instance), at least two elements are algebraically independent. One of the most interesting features of this paper is that all these results are deduced from only two general theorems, while in previous works on this subject, each single statement needed a specified proof [see for instance \textit{G. V. Chudnovsky}, Contributions to the theory of transcendental numbers (Math. Surv. Monogr. 19) (1984; Zbl 0594.10024)]. Here are the two main results. Let \(G={\mathbb{G}}_ a^{d_ 0}\times {\mathbb{G}}_ m^{d_ 1}\times G_ 2\) be a commutative algebraic group of dimension \(d=d_ 0+d_ 1+d_ 2\geq 1\), over a subfield K of \({\mathbb{C}}\). Let W be a subspace of \(T_ G({\mathbb{C}})\), of dimension \(n\geq 1\), and let V be a one-dimensional subspace of W. Assume that W is defined over \({\bar {\mathbb{Q}}}\). Further, let Y be a finitely generated subgroup of V, of rank \(\ell\) over \({\mathbb{Z}}\), such that \(\exp_ G Y\subset G(K)\). Assume furthermore that either (i) \(\ell (d-n)\geq \ell +2d_ 2+d_ 1\), with strict inequality in case \(d_ 0=n=1\), or (ii) \(Y\cap \ker \exp_ G\neq 0\) and \(2\ell (d-n)\geq \ell +2d_ 2+d_ 1\). Then the transcendence degree of K over \({\mathbb{Q}}\) is at least 2. Several new interesting consequences are derived. The proof involves the method of algebraic independence introduced by Gel'fond, an adaptation of Baker's method (for the case \(n\geq 2)\), and a refinement, due to Philippon, of a zero estimate of Masser-Wüstholz [\textit{P. Philippon}, Bull. Soc. Math. Fr. 114, 355-383 (1986)].
scientific article; zbMATH DE number 3983283
scientific article