论文标题
数字字段上的数字和标量扩展的参数几何形状
Parametric geometry of numbers over a number field and extension of scalars
论文作者
论文摘要
Schmidt和Summerer数量的参数几何涉及$ \ Mathbb {r}^n $中点的合理近似值。我们将此理论扩展到一个数字字段$ k $及其完成$ k_w $，以$ w $的位置$ w $，以便在$ k_w^n $中对$ k $上的近似值处理近似。结果，我们发现$ \ mathbb {q} $ of $ \ mathbb {r}^n $具有近似值的指数与$ k $ in $ k_w^n $中的$ k $上的概述相同。当$ w $在$ \ mathbb {q} $上具有相对学位的$ \ ell $时，我们将$ k $上的近似值与$ k_w^n $中的点$ \boldsymbolξ$与$ \ m athbb {q} $近似的近似值，以$ \ mathbb {q} $的近似为标量，其中$ d $是$ k $超过$ \ mathbb {q} $的度。结合BEL的结果，这使我们能够以$ \ Mathbb {r}^{3D} $定义在$ \ Mathbb {q} $上的$ 2D $的$ \ Mathbb {r}^{3d} $中构建代数曲线，其中包含相对于合理近似而言非常单数的点。
The parametric geometry of numbers of Schmidt and Summerer deals with rational approximation to points in $\mathbb{R}^n$. We extend this theory to a number field $K$ and its completion $K_w$ at a place $w$ in order to treat approximation over $K$ to points in $K_w^n$. As a consequence, we find that exponents of approximation over $\mathbb{Q}$ in $\mathbb{R}^n$ have the same spectrum as their generalizations over $K$ in $K_w^n$. When $w$ has relative degree one over a place $\ell$ of $\mathbb{Q}$, we further relate approximation over $K$ to a point $\boldsymbolξ$ in $K_w^n$, to approximation over $\mathbb{Q}$ to a point $Ξ$ in $\mathbb{Q}_\ell^{nd}$, obtained by extension of scalars, where $d$ is the degree of $K$ over $\mathbb{Q}$. By combination with a result of Bel, this allows us to construct algebraic curves in $\mathbb{R}^{3d}$ defined over $\mathbb{Q}$, of degree $2d$, containing points that are very singular with respect to rational approximation.
