FoCM 2014 conference

Workshop B7 - Symbolic Analysis

December 17, 16:00 ~ 16:25 - Room A11

## Computing the parameterized differential Galois group of a second-order linear differential equation with parameters

### Carlos E. Arreche

### North Carolina State University, USA - cearrech@ncsu.edu

Consider a linear differential equation $$\tfrac{\partial^2Y}{\partial x^2}+r_1\tfrac{\partial Y}{\partial x} +r_0Y=0,$$ where the coefficients $r_1,r_0\in\mathbb{C}(t_1,\dots , t_m,x)$. The parameterized Picard-Vessiot theory developed by Phyllis Cassidy and Michael Singer associates a differential Galois group $G$ to such an equation. In analogy with the classical Picard-Vessiot theory of Kolchin, $G$ measures the differential-algebraic relations amongst the solutions to the equation, with respect to $\tfrac{\partial}{\partial x}$ as well as $\tfrac{\partial}{\partial t_1},\dots,\tfrac{\partial}{\partial t_m}$.

Relying on earlier work by Thomas Dreyfus, I will describe a complete set of algorithms to compute $G$, and how these algorithms lead to a simple procedure to decide whether any of the solutions to the equation are differentially transcendental with respect to one or several of the parametric derivations $\tfrac{\partial}{\partial t_1},\dots,\tfrac{\partial}{\partial t_m}$.