## Points on subvarieties of tori.(English)Zbl 1040.11047

Wüstholz, Gisbert (ed.), A panorama in number theory or The view from Baker’s garden. Based on a conference in honor of Alan Baker’s 60th birthday, Zürich, Switzerland, 1999. Cambridge: Cambridge University Press (ISBN 0-521-80799-9/hbk). 214-230 (2002).
Let $$\mathbb{G}_{m}^{N}$$ be a $$N$$-dimensional torus over an algebraically closed field $$K$$ of characteristic zero and let $$\Gamma$$ be a finitely generated subgroup of $$\mathbb{G}_{m}^{N}(K)$$ with division subgroup $$\overline{\Gamma}$$. The author surveys results about the structure of $$X\cap\overline{\Gamma}$$ where $$X$$ is an algebraic subvariety of $$\mathbb{G}_{m}^{N}$$ defined over $$K$$. The most recent results are due to the author, W. M. Schmidt and H. P. Schlickewei, an example of which is the following. Let $$a_{1},\ldots, a_{N}$$ be elements in $$K^{\times}$$. A solution $$(x_{1},\ldots, x_{N})\in \overline{\Gamma}$$ of the equation $$a_{1}x_{1}+\cdots+a_{N}x_{N}=1$$ is called nondegenerate if $$\sum_{i\in I}a_{i}x_{i} \not=0$$ for any nonempty subset $$I$$ of $$\{1,\ldots,N\}$$. Then the set of nondegenerate solutions is at most $$e^{(6N)^{3N} (r+1)}$$, where $$r$$ is the rank of $$\Gamma$$. The author also provides a sketch of proof. The main tool is a quantitative version of the subspace theorem.
For the entire collection see [Zbl 0997.00017].

### MSC:

 11G35 Varieties over global fields 11D61 Exponential Diophantine equations 11J13 Simultaneous homogeneous approximation, linear forms 11G50 Heights 14G25 Global ground fields in algebraic geometry