Calculating differential Galois groups of parametrized differential equations, with applications to hypertranscendence

Abstract

The main motivation of our work is to create an efficient algorithm that decides hypertranscendence of solutions of linear differential equations, via the parameterized differential and Galois theories. To achieve this, we expand the representation theory of linear differential algebraic groups and develop new algorithms that calculate unipotent radicals of parameterized differential Galois groups for differential equations whose coefficients are rational functions. P. Berman and M.F. Singer presented an algorithm calculating the differential Galois group for differential equations without parameters whose differential operator is a composition of two completely reducible differential operators. We use their algorithm as a part of our algorithm. As a result, we find an effective criterion for the algebraic independence of the solutions of parameterized differential equations and all of their derivatives with respect to the parameter.

References in corpus (8)

• Tannakian categories, linear differential algebraic groups, and parameterized linear differential equations
• Hrushovski's Algorithm for Computing the Galois Group of a Linear Differential Equation
• Tannakian approach to linear differential algebraic groups
• On parameterized differential Galois extensions
• Computing the differential Galois group of a parameterized second-order linear differential equation
• Difference algebraic relations among solutions of linear differential equations
• A Jordan-Hoelder Theorem for Differential Algebraic Groups
• On the Computation of the Galois Group of Linear Difference Equations

Cited by in corpus (3)

• Galois groups for integrable and projectively integrable linear difference equations
• Finiteness properties of affine difference algebraic groups
• Calculating Galois groups of third order linear differential equations with parameters