Differential Galois Group Research Articles

A note on the inverse problem for finite differential Galois groups

In this paper we revisit the following inverse problem: given a curve invariant under an irreducible finite linear algebraic group, can we construct an ordinary linear differential equation whose Schwarz map parametrizes it? We present an algorithmic solution to this problem under the assumption that we are given the function field of the quotient curve. The result provides a generalization and an efficient implementation of the solution to the inverse problem exposed by van der Put et al. (2020). As an application, we show that there is no hypergeometric equation with projective differential Galois group isomorphic to H72, thus completing Beukers and Heckman’s answer (Beukers and Heckman, 1989) to the question of which irreducible finite subgroup of PSL3(ℂ) are the projective monodromy of a hypergeometric equation.

Dynamics and non-integrability of the double spring pendulum

This paper investigates the dynamics and integrability of the double spring pendulum, which has great importance in studying nonlinear dynamics, chaos, and bifurcations. Being a Hamiltonian system with three degrees of freedom, its analysis presents a significant challenge. To gain insight into the system’s dynamics, we employ various numerical methods, including Lyapunov exponents spectra, phase-parametric diagrams, and Poincaré cross-sections. The novelty of our work lies in the integration of these three numerical methods into one powerful tool. We provide a comprehensive understanding of the system’s dynamics by identifying parameter values or initial conditions that lead to hyper-chaotic, chaotic, quasi-periodic, and periodic motion, which is a novel contribution in the context of Hamiltonian systems. In the absence of gravitational potential, the system exhibits S1 symmetry, and the presence of an additional first integral was identified using Lyapunov exponents diagrams. We demonstrate the effective utilization of Lyapunov exponents as a potential indicator of first integrals and integrable dynamics. The numerical analysis is complemented by an analytical proof regarding the non-integrability of the system. This proof relies on the analysis of properties of the differential Galois group of variational equations along specific solutions of the system. To facilitate this analysis, we utilized a newly developed extension of the Kovacic algorithm specifically designed for fourth-order differential equations. Overall, our study sheds light on the intricate dynamics and integrability of the double spring pendulum, offering new insights and methodologies for further research in this field.

Regular singular differential equations and free proalgebraic groups

AbstractWe determine the differential Galois group of the family of all regular singular differential equations on the Riemann sphere. It is the free proalgebraic group on a set of cardinality .

Top on a smooth plane.

We investigate the dynamics of a sliding top that is a rigid body with an ideal sharp tip moving in a perfectly smooth horizontal plane, so no friction forces act on the body. We prove that this system is integrable only in two cases analogous to the Euler and Lagrange cases of the classical top problem. The cases with the constant gravity field with acceleration g≠0 and without external field g=0 are considered. The non-integrability proof for g≠0 is based on the fact that the equations of motion for the sliding top are a perturbation of the classical top equations of motion. We show that the integrability of the classical top is a necessary condition for the integrability of the sliding top. Among four integrable classical top cases, the corresponding two cases for the sliding top are also integrable, and for the two remaining cases, we prove their non-integrability by analyzing the differential Galois group of variational equations along a certain particular solution. In the absence of a constant gravitational field g=0, the integrability is much more difficult. First, we proved that if the sliding top problem is integrable, then the body is symmetric. In the proof, we applied the Ziglin theorem concerning the splitting of separatrices phenomenon. Then, we prove the non-integrability of the symmetric sliding top using the differential Galois group of variational equations except two the same as for g≠0 cases. The integrability of these cases is also preserved when we add to equations of motion a gyrostatic term.

Chaos and integrability of relativistic homogeneous potentials in curved space

Relativistic Hamiltonian systems of n degrees of freedom in static curved spaces are considered. The source of space-time curvature is a scalar potential V(q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$V(\\varvec{q})$$\\end{document}. In the limit of weak potential 2V(q)/mc2≪1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2V(\\varvec{q})/mc^2\\ll 1$$\\end{document}, and small momentum |p|/mc≪1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$|\\varvec{p} |/ mc\\ll 1$$\\end{document}, these systems transform into the corresponding non-relativistic flat Hamiltonian’s with the standard sum of kinetic energy plus potential V(q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$V(\\varvec{q})$$\\end{document}. We compare the dynamics of the classical and the corresponding relativistic curved counterparts on examples of important physical models: the Hénon–Heiles system, the Armbruster–Guckenheimer–Kim galactic system and swinging Atwood’s machine. Our main results are formulated for relativistic Hamiltonian systems with homogeneous potentials of non-zero integer degree k of homogeneity. First, we show that the integrability of a non-relativistic flat Hamiltonian with a homogeneous potential is a necessary condition for the integrability of its relativistic counterpart in curved space-time with the same homogeneous potential or with a non-homogeneous potential that the lowest homogenous part coincides with this homogeneous potential. Moreover, we formulate necessary integrability conditions for relativistic Hamiltonian systems with homogeneous potentials in curved space-time. These conditions were obtained from analysis of the differential Galois group of variational equations along a particular straight-line solution defined by a non-zero vector d\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{d}$$\\end{document} satisfying V′(d)=γd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$V'(\\varvec{d})=\\gamma \\varvec{d}$$\\end{document} for a certain γ≠0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma \ e 0$$\\end{document}. They are very strong: if the relativistic system is integrable in the Liouville sense, then either k=±2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k=\\pm 2$$\\end{document}, or all non-trivial eigenvalues of the re-scaled Hessian γ-1V′′(d)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma ^{-1}V''(\\varvec{d})$$\\end{document} are either 0, or 1. Certain integrable relativistic systems are presented.

Regular-singular connections on relative complex schemes

Deligne's celebrated "Riemann--Hilbert correspondence" relates representations of the fundamental group of a smooth complex algebraic variety and regular-singular integrable connections. In this work, we show how to arrive at a similar statement in the case of a smooth scheme $X$ over the spectrum of a ring $R=\mathbb C[[t_1,\ldots, t_r]]/I$. On one side of the correspondence we have representations on $R$-modules of the fundamental group of the special fibre, and on the other we have certain integrable $R$-connections admitting logarithmic models. The correspondence is then applied to give explicit examples of differential Galois groups of $\mathbb C[[t]]$--connections.

Nonintegrability of the Painlevé IV equation in the Liouville–Arnold sense and Stokes phenomena

AbstractIn this paper we study the integrability of the Hamiltonian system associated with the fourth Painlevé equation. We prove that one two‐parametric family of this Hamiltonian system is not integrable in the sense of the Liouville–Arnold theorem. Computing explicitly the Stokes matrices and the formal invariants of the second variational equations, we deduce that the connected component of the unit element of the corresponding differential Galois group is not Abelian. Thus the Morales–Ramis–Simó theory leads to a nonintegrable result. Moreover, combining the new result with our previous one we establish that for all values of the parameters for which the equation has a particular rational solution the corresponding Hamiltonian system is not integrable by meromorphic first integrals which are rational in t.

Wildly ramified rigid G2$G_2$‐local systems

AbstractIn earlier work of the author, rigid irregular connections with differential Galois group and whose slopes have numerator 1 were classified and new rigid connections were constructed. The same construction can be carried out for ‐adic local systems in the setting of positive characteristic. In this article, we provide the results that are needed to obtain the classification of wildly ramified rigid ‐local systems whose slopes have numerator 1. The overall strategy of the classification is very similar but the methods needed to obtain some invariants differ.

Subgroups of free proalgebraic groups and Matzat’s conjecture for function fields

We show that a closed finite index subgroup of a free proalgebraic group is itself a free proalgebraic group. As an application, we prove that the absolute differential Galois group of a one-variable function field over an algebraically closed characteristic zero field of infinite transcendence degree is a free proalgebraic group.

Normal forms in differential Galois theory for the classical groups

ABSTRACT Let G be a classical group of dimension d and let be differential indeterminates over a differential field F of characteristic zero with algebraically closed field of constants C. Further let be a generic element in the Lie algebra of G obtained from parametrizing a basis of with the indeterminates a . It is known (cf. [5]) that the differential Galois group of over is . In this paper we construct a differential field extension of such that the field of constants of is C, the differential Galois group of over is still the full group and is gauge equivalent over to a matrix in normal form which we introduced in [13]. We also consider specializations of the coefficients of .

Solutions of algebraic linear ordinary differential equations

A classical result of Klein states that, given a finite primitive group G⊆SL2(C), there exists a hypergeometric equation such that every second order LODE whose differential Galois group is isomorphic to G is projectively equivalent to the pullback by a rational map of this hypergeometric equation. In this paper, we generalize this result. We show that, given a finite primitive group G⊆SLn(C), there exist a positive integer d=d(G) and a standard equation such that every LODE whose differential Galois group is isomorphic to G is gauge equivalent, over a field extension F of degree d, to an equation projectively equivalent to the pullback by a map in F of this standard equation. For n=3, these standard equations can be chosen to be hypergeometric.

On the algebraic dependence of holonomic functions

We study the form of possible algebraic relations between functions satisfying linear differential equations. In particular, if f and g satisfy linear differential equations and are algebraically dependent, we give conditions on the differential Galois group associated to f guaranteeing that g is a polynomial in f. We apply this to hypergeometric functions and iterated integrals.

Computing the Lie algebra of the differential Galois group: The reducible case

In this paper, we explain how to compute the Lie algebra of the differential Galois group of a reducible linear differential system. We achieve this by showing how to transform a block-triangular linear differential system into a Kolchin-Kovacic reduced form. We combine this with other reduction results to propose a general algorithm for computing a reduced form of a general linear differential system. In particular, this provides directly the Lie algebra of the differential Galois group without an a priori computation of this Galois group.

The differential Galois group of the maximal prounipotent extension is free

Let F F be a characteristic zero differential field with algebraically closed constant field, and consider the compositum F u F_u of all Picard–Vessiot extensions of F F with unipotent differential Galois group. We prove that the group of F F differential automorphisms of F u F_u is a free prounipotent group.

Computing differential Galois groups of second-order linear q-difference equations

We apply the differential Galois theory for difference equations developed by Hardouin and Singer to compute the differential Galois group for a second-order linear q-difference equation with rational function coefficients. This Galois group encodes the possible polynomial differential relations among the solutions of the equation. We apply our results to compute the differential Galois groups of several concrete q-difference equations, including for the colored Jones polynomial of a certain knot.

Free differential Galois groups

We study the structure of the absolute differential Galois group of a rational function field over an algebraically closed field of characteristic zero. In particular, we relate the behavior of differential embedding problems to the condition that the absolute differential Galois group is free as a proalgebraic group. Building on this, we prove Matzat’s freeness conjecture in the case that the field of constants is algebraically closed of countably infinite transcendence degree over Q \mathbb {Q} . This is the first known case of the twenty year old conjecture.

The differential Galois group of the rational function field

We determine the absolute differential Galois group of the field C(x) of rational functions: It is the free proalgebraic group on a set of cardinality |C|. This solves a longstanding open problem posed by B.H. Matzat. For the proof we develop a new characterization of free proalgebraic groups in terms of split embedding problems, and we use patching techniques in order to solve a very general class of differential embedding problems. Our result about C(x) also applies to rational function fields over more general fields of coefficients.

Difference Galois groups under specialization

We present a difference analogue of a result given by Hrushovski on differential Galois groups under specialization. Let k k be an algebraically closed field of characteristic zero and let X \mathbb {X} be an irreducible affine algebraic variety over k k . Consider the linear difference equation σ ( Y ) = A Y , \begin{equation*} \sigma (Y)=AY, \end{equation*} where A ∈ G L n ( k ( X ) ( x ) ) A\in \mathrm {GL}_n(k(\mathbb {X})(x)) and σ \sigma is the shift operator σ ( x ) = x + 1 \sigma (x)=x+1 . Assume that the Galois group G G of the above equation over k ( X ) ¯ ( x ) \overline {k(\mathbb {X})}(x) is defined over k ( X ) k(\mathbb {X}) , i.e., the vanishing ideal of G G is generated by a finite set S ⊂ k ( X ) [ X , 1 / det ( X ) ] S\subset k(\mathbb {X})[X,1/\det (X)] . For a c ∈ X {\mathbf {c}}\in \mathbb {X} , denote by v c v_{{\mathbf {c}}} the map from k [ X ] k[\mathbb {X}] to k k given by v c ( f ) = f ( c ) v_{{\mathbf {c}}}(f)=f({\mathbf {c}}) for any f ∈ k [ X ] f\in k[\mathbb {X}] . We prove that the set of c ∈ X {\mathbf {c}}\in \mathbb {X} satisfying that v c ( A ) v_{\mathbf {c}}(A) and v c ( S ) v_{\mathbf {c}}(S) are well-defined and the affine variety in G L n ( k ) \mathrm {GL}_n(k) defined by v c ( S ) v_{{\mathbf {c}}}(S) is the Galois group of σ ( Y ) = v c ( A ) Y \sigma (Y)=v_{{\mathbf {c}}}(A)Y over k ( x ) k(x) is Zariski dense in X \mathbb {X} . We apply our result to van der Put-Singer’s conjecture which asserts that an algebraic subgroup G G of G L n ( k ) \mathrm {GL}_n(k) is the Galois group of a linear difference equation over k ( x ) k(x) if and only if the quotient G / G ∘ G/G^\circ by the identity component is cyclic. We show that if van der Put-Singer’s conjecture is true for k = C k=\mathbb {C} , then it will be true for any algebraically closed field k k of characteristic zero.

On the Structure of Affine Flat Group Schemes Over Discrete Valuation Rings, II

Abstract In the first part of this work [ 12], we studied affine group schemes over a discrete valuation ring (DVR) by means of Neron blowups. We also showed how to apply these findings to throw light on the group schemes coming from Tannakian categories of $\mathcal{D}$-modules. In the present work, we follow up this theme. We show that a certain class of affine group schemes of “infinite type,” Neron blowups of formal subgroups, are quite typical. We also explain how these group schemes appear naturally in Tannakian categories of $\mathcal{D}$-modules. To conclude, we isolate a Tannakian property of affine group schemes, named prudence, which allows one to verify if the underlying ring of functions is a free module over the base ring. This is then successfully applied to obtain a general result on the structure of differential Galois groups over complete DVRs.

Reduced group schemes as iterative differential Galois groups

This article is on the inverse Galois problem in Galois theory of linear iterative differential equations in positive characteristic. We show that it has an affirmative answer for reduced algebraic group schemes over any iterative differential field which is finitely generated over its algebraically closed field of constants.We also introduce the notion of equivalence of iterative derivations on a given field—a condition which implies that the inverse Galois problem over equivalent iterative derivations are equivalent.