On parameterized differential Galois extensions

TL;DR

This paper establishes new existence results for parameterized differential Galois extensions, extending previous work from ordinary differential equations to partial differential equations, and generalizing key theorems in the field.

Contribution

It generalizes existing results on parameterized Picard-Vessiot extensions from ODEs to PDEs and proves new existence theorems for parameterized strongly normal extensions.

Findings

Proved existence of parameterized strongly normal extensions for logarithmic equations.

Extended results from ODE to PDE cases in differential Galois theory.

Generalized key theorems to broader classes of differential equations.

Abstract

We prove some existence results on parameterized strongly normal extensions for logarithmic equations. We generalize a result in [Wibmer, Existence of d-parameterized Picard-Vessiot extensions over fields with algebraically closed constants, J. Algebra, 361, 2012]. We also consider an extension of the results in [Kamensky and Pillay, Interpretations and differential Galois extensions, Preprint 2014] from the ODE case to the parameterized PDE case.