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

TL;DR

This paper develops algorithms to compute differential Galois groups of parametrized linear differential equations, enabling effective criteria for hypertranscendence and algebraic independence of solutions and their derivatives.

Contribution

It extends the representation theory of linear differential algebraic groups and introduces new algorithms for calculating unipotent radicals of parameterized differential Galois groups.

Findings

Algorithm for calculating unipotent radicals of parameterized Galois groups.

Effective criterion for algebraic independence of solutions and derivatives.

Integration of existing algorithms for non-parameterized cases.

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.