Galois groups of differential equations

This project aims to further the understanding of algebraic properties of solutions of linear differential equations. The main focus is on linear differential equations whose coefficients are rational functions. Linear differential equations are ubiquitous in science and engineering. To perform symbolic computations with the solutions of linear differential equations it is essential to understand the algebraic relations among the solutions. The algebraic relations among the solutions of a given linear differential equation are governed by a linear algebraic group, called the differential Galois of the differential equation. All differential Galois groups of all linear differential equations with rational function coefficients fit together to form the absolute differential Galois group of the field of rational functions. The main goal of this project is to establish an explicit description of this group. Indeed, following B.H. Matzat we conjecture that this group is a free proalgebraic group on a set whose cardinality agrees with the cardinality of the field of coefficients of the rational functions. The absolute differential Galois group of a differential field can be seen a differential analog of the absolute Galois group of a field. According to a Theorem of A. Douady, F. Pop and D. Harbater, the absolute Galois group of the field of rational functions is a free profinite group. Matzat`s conjecture would generalize this theorem. Important methods for this project include patching techniques and embedding problems. Out plan to prove Matzat`s conjecture has two steps. Firstly we want to characterize free proalgebraic groups in terms of embedding problems and secondly we would like to show that the absolute differential Galois group satisfies this characterization. For the second step patching techniques will be used.

This project helped to further our understanding of algebraic properties of solutions of linear differential equations. The main focus was on linear differential equations whose coefficients are rational functions. Linear differential equations are ubiquitous in science and engineering. To perform symbolic computations with the solutions of linear differential equations, it is essential to understand the algebraic relations among the solutions. The algebraic relations among the solutions of a given linear differential equation are governed by a linear algebraic group, called the differential Galois of the differential equation. All differential Galois groups of all linear differential equations with rational function coefficients fit together to form the absolute differential Galois group of the field of rational functions. The main goal of this project was to establish an explicit description of this group. Indeed, we were able to prove a conjecture of Professor B.H. Matzat: The absolute differential Galois of the field of rational function is a free proalgebraic group on a set whose cardinality agrees with the cardinality of the field of coefficients of the rational functions. The absolute differential Galois group of a differential field can be seen as a differential analog of the absolute Galois group of a field. According to a Theorem of A. Douady, F. Pop and D. Harbater, the absolute Galois group of the field of rational functions is a free profinite group. Matzat's conjecture generalizes this theorem. Our solution of Matzat's conjecture has far-reaching applications in differential Galois theory. For example, Matzat's conjecture implies a quantitative sharpening of the solution of the inverse problem in differential Galois theory. The solution of the inverse problem states that every linear algebraic group occurs as the differential Galois of a linear differential equation with rational function coefficients. Matzat's conjecture implies the stronger statement that every non-trivial linear algebraic group occurs many times, indeed, it occurs as many times as there are elements in the coefficient field of the rational functions. Important methods that were used and further developed in this project include patching techniques, embedding problems and specialization results for differential Galois groups. Our proof of Matzat's conjecture has two steps. Firstly, we characterize free proalgebraic groups in terms of embedding problems. Secondly, we show that the absolute differential Galois group of the rational function field satisfies this characterization. Depending on the cardinality and the transcendence degree of the field of coefficients of the rational functions, the second step is easier or harder. The most difficult case arises when the coefficient field has finite transcendence degree. For this case we developed specialization results for differential Galois groups that are of interest beyond this project.