Picard-Vessiot Extensions For Unipotent Algebraic Groups

TL;DR

This paper constructs specific Picard-Vessiot extensions over differential fields with unipotent algebraic group Galois groups, providing a method to compute associated differential operators without requiring algebraically closed constants.

Contribution

It introduces a construction of Picard-Vessiot extensions with unipotent Galois groups over fields with arbitrary constants and offers a procedure to compute their differential operators.

Findings

Constructed Picard-Vessiot extensions with unipotent Galois groups

Provided a method to compute linear differential operators for these extensions

Did not require algebraically closed constant fields

Abstract

Let F be a differential field of characteristic zero. In this article, we construct Picard-Vessiot extensions of F whose differential Galois group is isomorphic to the full unipotent subgroup of the upper triangular group defined over the field of constants of F. We will also give a procedure to compute linear differential operators for our Picard-Vessiot extensions. We do not require the condition that the field of constants be algebraically closed.