Hrushovski's Algorithm for Computing the Galois Group of a Linear Differential Equation

We present a detailed and simplified version of Hrushovski's algorithm that determines the Galois group of a linear differential equation. There are three major ingredients in this algorithm. The first is to look for a degree bound for proto-Galois groups, which enables one to compute one of them. The second is to determine the identity component of the Galois group that is the pullback of a torus to the proto-Galois group. The third is to recover the Galois group from its identity component and a finite Galois group.Comment: 27 page

A Reduced Form for Linear Differential Systems and its Application to Integrability of Hamiltonian Systems

Let $[A]: Y'=AY$ with $A\in \mathrm{M}_n (k)$ be a differential linear system. We say that a matrix $R\in {\cal M}_{n}(\bar{k})$ is a {\em reduced form} of $[A]$ if $R\in \mathfrak{g}(\bar{k})$ and there exists $P\in GL_n (\bar{k})$ such that $R=P^{-1}(AP-P')\in \mathfrak{g}(\bar{k})$. Such a form is often the sparsest possible attainable through gauge transformations without introducing new transcendants. In this article, we discuss how to compute reduced forms of some symplectic differential systems, arising as variational equations of hamiltonian systems. We use this to give an effective form of the Morales-Ramis theorem on (non)-integrability of Hamiltonian systems.Comment: 28 page

On the Complexity of Computing Galois Groups of Differential Equations

The differential Galois group is an analogue for a linear differential equation of the classical Galois group for a polynomial equation. An important application of the differential Galois group is that a linear differential equation can be solved by integrals, exponentials and algebraic functions if and only if the connected component of its differential Galois group is solvable. Computing the differential Galois groups would help us determine the existence of the solutions expressed in terms of elementary functions (integrals, exponentials and algebraic functions) and understand the algebraic relations among the solutions. Hrushovski first proposed an algorithm for computing the differential Galois group of a general linear differential equation. Recently, Feng approached finding a complexity bound of the algorithm, which is the degree bound of the polynomials used in the first step of the algorithm for finding a proto-Galois group. The bound given by Feng is quintuply exponential in the order n of the differential equation. The complexity, in the worst case, of computing a Gröbner basis is doubly exponential in the number of variables. Feng chose to represent the radical of the ideal generated by the defining equations of a proto-Galois group by its Gröbner basis. Hence, a double-exponential degree bound for computing Gröbner bases was involved when Feng derived the complexity bound of computing a proto-Galois group. Triangular decomposition provides an alternative method for representing the radical of an ideal. It represents the radical of an ideal by the triangular sets instead of its generators. The first step of Hrushovski\u27s algorithm is to find a proto-Galois group which can be used further to find the differential Galois group. So it is important to analyze the complexity for finding a proto-Galois group. We represent the radical of the ideal generated by the defining equations of a proto-Galois group using the triangular sets instead of the generating sets. We apply Szántó\u27s modified Wu-Ritt type decomposition algorithm and make use of the numerical bound for Szántó\u27s algorithm to adapt to the complexity analysis of Hrushovski\u27s algorithm. We present a triple exponential degree upper bound for finding a proto-Galois group in the first step of Hrushovski\u27s algorithm

An algorithmic approach to the differential Galois theory of second-order linear differential equations with differential parameters

We present algorithms to compute the differential Galois group G associated via the parameterized Picard-Vessiot theory to a parameterized second-order linear differential equation with respect to d/dx, with coefficients in the field of rational functions F(x) over a differential field F, where we think of the derivations on F as being derivations with respect to parameters. We build on an earlier procedure, developed by Dreyfus, that computes G when the equation is unimodular, assuming either that G is reductive, or else that its maximal reductive quotient is differentially constant. We first show how to modify the space of parametric derivations to reduce the computation of the unipotent radical of G to the case when the reductive quotient is differentially constant in the unimodular case. For non-unimodular equations, we reinterpret a classical change-of-variables procedure in Galois-theoretic terms in order to reduce the computation of G to the computation of an associated unimodular differential Galois group H. We establish a parameterized version of the Kolchin-Ostrowski theorem and apply it to give more direct proofs than those found in the literature of the fact that the required computations can be performed effectively. We then extract from these algorithms a complete set of criteria to decide whether any of the solutions to a parameterized second-order linear differential equation is differentially transcendental with respect to the parametric derivations. We give various examples of computation and some applications to differential transcendenc

Galois Groups of Differential Equations and Representing Algebraic Sets

The algebraic framework for capturing properties of solution sets of differential equations was formally introduced by Ritt and Kolchin. As a parallel to the classical Galois groups of polynomial equations, they devised the notion of a differential Galois group for a linear differential equation. Just as solvability of a polynomial equation by radicals is linked to the equation’s Galois group, so too is the ability to express the solution to a linear differential equation in closed form linked to the equation’s differential Galois group. It is thus useful even outside of mathematics to be able to compute and represent these differential Galois groups, which can be realized as linear algebraic groups; indeed, many algorithms have been written for this purpose. The most general of these is Hrushovski\u27s algorithm and so its complexity is of great interest. A key step of the algorithm is the computation of a group called a proto-Galois group, which contains the differential Galois group. As a proto-Galois group is an algebraic set and there are various ways to represent an algebraic set, a natural matter to investigate in this regard is which representation(s) are expected to be the smallest. Some typical representations of algebraic sets are equations (that have the given algebraic set as their common solutions) and, for the corresponding radical ideal, Groebner bases or triangular sets. In computing any of these representations, it can be helpful to have a degree bound on the polynomials they will feature based on the given differential equation. Feng gave such a bound for a Groebner basis for a proto-Galois group\u27s radical ideal in terms of the size of the coefficient matrix. We first discuss an improvement of this bound achieved by focusing on equations that define such a group instead of its corresponding ideal. This bound also produces a smaller degree bound for Groebner bases than the one Feng obtained. Recent work by M. Sun shows that Feng\u27s bound can also be improved by replacing Feng\u27s uses of Groebner bases by triangular sets. Sun\u27s bound relies on results on the complexity of triangular representations of algebraic sets, results that we shall present and that more generally suggest using triangular sets in place of Groebner bases to potentially reduce complexity