TU Berlin

Fachgebiet Algorithmische AlgebraPublications

Page Content

to Navigation

There is no English translation for this web page.

Search for Publication

Search for publications

All Publications

Nonarchimedean integral geometry
Citation key BKL-Nonarchimedean-integral-geometry
Author Peter Bürgisser and Avinash Kulkarni and Antonio Lerario
Year 2022
Month 06
Abstract Let K be a nonarchimedean local field of characteristic zero with valuation ring R, for instance, K=ℚp and R=ℤp. We introduce the concept of an R-structure on a K-analytic manifold, which is akin to the notion of a Riemannian metric on a smooth manifold. An R-structure determines a norm and volume form on the tangent spaces of a K-analytic manifold, which allows one to integrate real valued functions on the manifold. We prove such integrals satisfy a nonarchimedean version of Sard's lemma and the coarea formula. We also consider R-structures on K-analytic groups and homogeneous K-analytic spaces that are compatible with the group operations. In this framework, we prove a general integral geometric formula 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. As a first application, we compute the volume of special Schubert varieties and we outline the construction of a probabilistic nonarchimedean Schubert Calculus. As a second application we bound, and in some cases exactly determine, the expected number of zeros in K of random fewnomial systems.
Link to publication Download Bibtex entry


Quick Access

Schnellnavigation zur Seite über Nummerneingabe