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].


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