On the finite inverse problem in iterative differential Galois theory

TL;DR

This paper explores the structure of finite group schemes in iterative differential Galois theory over fields of positive characteristic, revealing which schemes can occur as Galois groups and describing their properties.

Contribution

It characterizes the finite group schemes that can appear as Galois groups over certain ID-fields, expanding understanding of the inverse problem in positive characteristic.

Findings

Classification of finite group schemes in the Galois correspondence

Description of all occurring finite group schemes for large classes of ID-fields

Insights into the structure of nonreduced and infinitesimal Galois group schemes

Abstract

In positive characteristic, nearly all Picard-Vessiot extensions are inseparable over some intermediate iterative differential extensions. In the Galois correspondence, these intermediate fields correspond to nonreduced subgroup schemes of the Galois group scheme. Moreover, the Galois group scheme itself may be nonreduced, or even infinitesimal. In this article, we investigate which finite group schemes occur as iterative differential Galois group schemes over a given ID-field. For a large class of ID-fields, we give a description of all occuring finite group schemes.