Nonarchimedean integral geometry

Abstract

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.