论文标题
非建筑积分几何
Nonarchimedean integral geometry
论文作者
论文摘要
令$ k $为非架构的特征零本地字段，带有估值环$ r $，例如，$ k = \ mathbb {q} _p $和$ r = \ mathbb {z} _p $。我们证明了$ K $分析群和均匀的$ k $分析空间的一般积分几何公式，类似于对真实的相应结果。这概括了Kulkarni和Lerario最近发现的投影空间的$ p $ -adic积分几何公式，例如，grassmannians的环境。基于这一点，我们概述了非建筑概率舒伯特演算的构建。为此，我们通过位置向量（一个非构造矢量的非一切集类似物）来表征$ k^n $的两个子空间的相对位置，并且我们研究了随机统一子空间的位置向量的概率分布。然后，我们使用它来计算$ k $以上的特殊舒伯特品种的体积。作为一般积分几何公式的第二次应用，我们启动了在非架构磁场上的随机少量典型系统的研究，边界，并且在某些情况下确切地确定了此类随机系统的预期零数。
Let $K$ be a nonarchimedean local field of characteristic zero with valuation ring $R$, for instance, $K=\mathbb{Q}_p$ and $R=\mathbb{Z}_p$. We prove a general integral geometric formula for $K$-analytic groups and homogeneous $K$-analytic spaces, analogous to the corresponding result over the reals. This generalizes the $p$-adic integral geometric formula for projective spaces recently discovered by Kulkarni and Lerario, e.g., to the setting of Grassmannians. Based on this, we outline the construction of a nonarchimedean probabilistic Schubert Calculus. For this purpose, we characterize the relative position of two subspaces of $K^n$ by a position vector, a nonarchimedean analogue of the notion of principal angles, and we study the probability distribution of the position vector for random uniform subspaces. We then use this to compute the volume of special Schubert varieties over $K$. As a second application of the general integral geometry formula, we initiate the study of random fewnomial systems over nonarchimedean fields, bounding, and in some cases exactly determining, the expected number of zeros of such random systems.
