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**Transcendental numbers and functions of several variables.**
*(English)*
Zbl 0791.11032

Gouvêa, Fernando Q. (ed.) et al., Advances in number theory. The proceedings of the third conference of the Canadian Number Theory Association, held at Queen’s University, Kingston, Canada, August 18-24, 1991. Oxford: Clarendon Press. 67-80 (1993).

This paper presents a survey of recent applications of the theorem of the algebraic subgroup [the author, New advances in transcendence theory 1986, 375-398 (1988; Zbl 0659.10035)] together with a new proof of this result using interpolation determinants.

The first two applications concern density questions. One is the existence of finitely generated subgroups of the multiplicative group \(k^ \times\) of a number field \(k\) whose image is dense in \((k\otimes \mathbb{R})^{\times}\), together with an answer to the question of J.-J. Sansuc about the smallest rank of such a subgroup [D. Roy, Invent. Math. 109, 547-556 (1992; Zbl 0780.11060)]. The other is a step towards a conjecture of B. Mazur. It says that if the Mordell-Weil group \(A(\mathbb{Q})\) of a simple Abelian variety \(A\) defined over \(\mathbb{Q}\) is sufficiently large then the topological closure of \(A(\mathbb{Q})\) in \(A(\mathbb{R})\) contains the neutral component of \(A(\mathbb{R})\).

The author also gives four different ways to recover Baker’s theorem from the algebraic subgroup theorem. They are dual by pairs associated to A. Baker’s method and N. Hirata’s method. The duality in question is explained in the paper and is reflected by the fact that the corresponding interpolation matrices are transposed of one another [see the author, J. Anal. Math. 56, 231-254, 255-279 (1991; Zbl 0742.11035 and Zbl 0742.11036)].

The last application is motivated by Leopold’s conjecture and concerns an arithmetic lower bound for the rank of matrices whose coefficients are linear combinations of logarithms of algebraic numbers with algebraic coefficients [the reviewer, J. Number Theory 41, 22-47 (1992; Zbl 0763.11030)]. There is also mention of a generalization of this type of result in the context of algebraic groups [D. Roy and M. Waldschmidt, Autour du théorème du sous-groupe algébrique, Can. Math. Bull. 36, 358-367 (1993)].

For the entire collection see [Zbl 0773.00021].

The first two applications concern density questions. One is the existence of finitely generated subgroups of the multiplicative group \(k^ \times\) of a number field \(k\) whose image is dense in \((k\otimes \mathbb{R})^{\times}\), together with an answer to the question of J.-J. Sansuc about the smallest rank of such a subgroup [D. Roy, Invent. Math. 109, 547-556 (1992; Zbl 0780.11060)]. The other is a step towards a conjecture of B. Mazur. It says that if the Mordell-Weil group \(A(\mathbb{Q})\) of a simple Abelian variety \(A\) defined over \(\mathbb{Q}\) is sufficiently large then the topological closure of \(A(\mathbb{Q})\) in \(A(\mathbb{R})\) contains the neutral component of \(A(\mathbb{R})\).

The author also gives four different ways to recover Baker’s theorem from the algebraic subgroup theorem. They are dual by pairs associated to A. Baker’s method and N. Hirata’s method. The duality in question is explained in the paper and is reflected by the fact that the corresponding interpolation matrices are transposed of one another [see the author, J. Anal. Math. 56, 231-254, 255-279 (1991; Zbl 0742.11035 and Zbl 0742.11036)].

The last application is motivated by Leopold’s conjecture and concerns an arithmetic lower bound for the rank of matrices whose coefficients are linear combinations of logarithms of algebraic numbers with algebraic coefficients [the reviewer, J. Number Theory 41, 22-47 (1992; Zbl 0763.11030)]. There is also mention of a generalization of this type of result in the context of algebraic groups [D. Roy and M. Waldschmidt, Autour du théorème du sous-groupe algébrique, Can. Math. Bull. 36, 358-367 (1993)].

For the entire collection see [Zbl 0773.00021].

Reviewer: D.Roy (Ottawa)