论文标题
通过半简单元素的有限产生：定量结果
Bounded Generation by semi-simple elements: quantitative results
论文作者
论文摘要
我们证明，对于一个数字字段$ f $，集合$σ\ subset \ mathbb {a} _f^n $具有纯粹指数参数化的分布，例如，在级别的尺寸时，在最多可靠的尺寸时，由半简单（对数）元素限制生成的矩阵（对数（对角线）元素。结果，一个人在特征零的字段$ k $上获得了线性组$γ\ subset \ mathrm {gl} _n（k）$，并且仅当其有限地生成并且其Zariski关闭的Zariski关闭的连接组件是一个纯粹的时指数的参数。我们的结果是通过关键的不平等值获得的，涉及纯指数参数化的最小$ m $ tuplass的高度。我们证明的主要成分是Evertse加强$ S $ UNIT方程定理。
We prove that for a number field $F$, the distribution of the points of a set $Σ\subset \mathbb{A}_F^n$ with a purely exponential parametrization, for example a set of matrices boundedly generated by semi-simple (diagonalizable) elements, is of at most logarithmic size when ordered by height. As a consequence, one obtains that a linear group $Γ\subset \mathrm{GL}_n(K)$ over a field $K$ of characteristic zero admits a purely exponential parametrization if and only if it is finitely generated and the connected component of its Zariski closure is a torus. Our results are obtained via a key inequality about the heights of minimal $m$-tuples for purely exponential parametrizations. One main ingredient of our proof is Evertse's strengthening of the $S$-Unit Equation Theorem.
