Nonarchimedean integral geometry

TL;DR

This paper develops a nonarchimedean integral geometric formula for $K$-analytic spaces, enabling probabilistic Schubert calculus and analysis of random fewnomial systems over fields like $ ext{Q}_p$, extending classical real results.

Contribution

It generalizes integral geometry to nonarchimedean fields, introduces a nonarchimedean notion of principal angles, and applies these to probabilistic Schubert calculus and zero-counting of random systems.

Findings

Derived a nonarchimedean integral geometric formula for $K$-analytic spaces.

Characterized the relative position of subspaces via a nonarchimedean analogue of principal angles.

Computed volumes of Schubert varieties and bounded the expected zeros of random fewnomial systems.

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.