Unipotent differential algebraic groups as parameterized differential Galois groups

TL;DR

This paper characterizes unipotent differential algebraic groups as parameterized differential Galois groups, providing criteria based on differential type and methods to determine and compute these groups for certain differential equations.

Contribution

It establishes a characterization of unipotent LDAGs as PPV Galois groups via differential type 0 and offers procedures for their determination and calculation.

Findings

Unipotent LDAGs are PPV Galois groups if and only if they have differential type 0.

Provides a method to determine if a differential equation's Galois group belongs to this class.

Shows how to compute the Galois group when it has differential type 0.

Abstract

We deal with aspects of the direct and inverse problems in parameterized Picard-Vessiot (PPV) theory. It is known that, for certain fields, a linear differential algebraic group (LDAG) G is a PPV Galois group over these fields if and only if G contains a Kolchin-dense finitely generated group. We show that, for a class of LDAGs G, including unipotent groups, G is such a group if and only if it has differential type 0. We give a procedure to determine if a parameterized linear differential equation has a PPV Galois group in this class and show how one can calculate the PPV Galois group of a parameterized linear differential equation if its Galois group has differential type 0.