Galois Theory Research Articles

Singularity analysis of a generalised Lagrange system

Abstract We study the integrability of a Hamiltonian system proposed recently by Maciejewski and Przybylska, and which constitutes an extension of one studied (and integrated) by Lagrange. While the previous authors used the differential Galois theory formalism in order to study the integrability of this extended Hamiltonian, we approach the problem from the point of view of singularity analysis. For all values of the parameter (integer in the study of Maciejewski and Przybylska and rational in ours) we are able to show that the system does not possess the Painleve property, with the exception of the case of the harmonic oscillator. The singularity structure of the Lagrange case is analysed in detail and commented upon.

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.

A differential Galois approach to path integrals

We point out the relevance of the differential Galois theory of linear differential equations for the exact semiclassical computations in path integrals in quantum mechanics. The main tool will be a necessary condition for complete integrability of classical Hamiltonian systems obtained by Ramis and myself [Morales-Ruiz and Ramis, Methods Appl. Anal. 8, 33–96 (2001); see also Morales-Ruiz, in Differential Galois Theory and Non-Integrability of Hamiltonian Systems, Modern Birkhäuser Classics (Springer, Basel, 1999)]; if a finite dimensional complex analytical Hamiltonian system is completely integrable with meromorphic first integrals, then the identity component of the Galois group of the variational equation around any integral curve must be abelian. A corollary of this result is that, for finite dimensional integrable Hamiltonian systems, the semiclassical approach is computable in a closed form in the framework of the differential Galois theory. This explains in a very precise way the success of quantum semiclassical computations for integrable Hamiltonian systems.

Restriction and extension of partial actions

Given a partial action α=(Ag,αg)g∈G of a connected groupoid G on a ring A and an object x of G, the isotropy group G(x) acts partially on the ideal Ax of A by the restriction of α. In this paper we investigate the following reverse question: under which conditions a partial group action of G(x) on an ideal of A can be extended to a partial groupoid action of G on A? The globalization problem and some applications to the Morita and Galois theories are also considered.

Walks in the quarter plane: Genus zero case

We use Galois theory of difference equations to study the nature of the generating series of (weighted) walks in the quarter plane with genus zero kernel curve. Using this approach, we prove that the generating series do not satisfy any nontrivial (possibly nonlinear) algebraic differential equation with rational coefficients.

Is Mathematics Unreasonably Effective?

ABSTRACT Many mathematicians, physicists, and philosophers have suggested that the fact that mathematics—an a priori discipline informed substantially by aesthetic considerations—can be applied to natural science is mysterious. This paper sharpens and responds to a challenge to this effect. I argue that the aesthetic considerations used to evaluate and motivate mathematics are much more closely connected with the physical world than one might presume, and (with reference to case studies within Galois theory and probabilistic number theory) I show that they are correlated with generally recognised theoretical virtues, such as explanatory depth, unifying power, fruitfulness, and importance.

Universal Central Extensions of Internal Crossed Modules via the Non-abelian Tensor Product

In the context of internal crossed modules over a fixed base object in a given semi-abelian category, we use the non-abelian tensor product in order to prove that an object is perfect (in an appropriate sense) if and only if it admits a universal central extension. This extends results of Brown-Loday (in the case of groups) and Edalatzadeh (in the case of Lie algebras). Our aim is to explain how those results can be understood in terms of categorical Galois theory: Edalatzadeh's interpretation in terms of quasi-pointed categories applies, but a more straightforward approach based on the theory developed in a pointed setting by Casas and the second author works as well.

The Lie group approach to solving differential equations

Certain ideas recur in many areas of mathematics. One example is groups of symmetries, which appear in the Galois theory of equations and in Lie groups. Lie groups are of great value in physics, where Noether’s theorem enables us to derive a conservation law for every case in which a function known as the Lagrangian is invariant under a one-parameter Lie group. The importance of this approach can be seen from the fact that the laws of the conservation of energy, linear momentum and angular momentum are all outcomes of Noether’s theorem, though they can of course be derived by simpler methods. The full power of Noether’s approach is shown in its applications to quantum field theory, where it can be used to find conserved currents and charges.

Integrability Analysis of the Stretch\u2013Twist\u2013Fold Flow

We study the integrability of an eight-parameter family of three-dimensional spherically confined steady Stokes flows introduced by Bajer and Moffatt. This volume-preserving flow was constructed to model the stretch–twist–fold mechanism of the fast dynamo magnetohydrodynamical model. In particular we obtain a complete classification of cases when the system admits an additional Darboux polynomial of degree one. All but one such case are integrable, and first integrals are presented in the paper. The case when the system admits an additional Darboux polynomial of degree one but is not evidently integrable is investigated by methods of differential Galois theory. It is proved that the four-parameter family contained in this case is not integrable in the Jacobi sense, i.e. it does not admit a meromorphic first integral. Moreover, we investigate the integrability of other four-parameter {textit{STF}} systems using the same methods. We distinguish all the cases when the system satisfies necessary conditions for integrability obtained from an analysis of the differential Galois group of variational equations.

Ordinary primes in Hilbert modular varieties

A well-known conjecture, often attributed to Serre, asserts that any motive over any number field has infinitely many ordinary reductions (in the sense that the Newton polygon coincides with the Hodge polygon). In the case of Hilbert modular cuspforms $f$ of parallel weight $(2,\ldots ,2)$, we show how to produce more ordinary primes by using the Sato–Tate equidistribution and combining it with the Galois theory of the Hecke field. Under the assumption of stronger forms of Sato–Tate equidistribution, we get stronger (but conditional) results. In the case of higher weights, we formulate the ordinariness conjecture for submotives of the intersection cohomology of proper algebraic varieties with motivic coefficients, and verify it for the motives whose $\ell$-adic Galois realisations are abelian on a finite-index subgroup. We get some results for Hilbert cuspforms of weight $(3,\ldots ,3)$, weaker than those for $(2,\ldots ,2)$.

Non-integrability on AdS3 supergravity backgrounds

We investigate classical integrability on two recently discovered classes of back- grounds in massive IIA supergravity. These vacua are of the form AdS3 × S2 × ℝ × CY2, they preserve small mathcal{N} = (0, 4) supersymmetry and are associated with D8−D6−D4−D2 Hanany-Witten brane set-ups. We choose an appropriate string embedding and use differential Galois theory on its associated Hamiltonian system, intending to produce the conditions under which Liouvillian solutions may occur. By constraining the parameters of the system according to the consistency of the associate brane set-ups we prove that no such conditions exist, yielding the complete non-integrability of these vacua. That is, up to the trivial cases where the background reduces to the Abelian and non-Abelian T-dual of AdS3 × S3 × T4.

Расширения Инабы полных полей характеристики 0

В статье изучаются p--расширения полных дискретно нормированных полей смешанной характеристики, где p-характеристика поля вычетов рассматриваемого поля. Известно, что любое вполне разветвленное расширение Галуа степени p-с немаксимальным скачком ветвления может быть задано уравнением Артина-Шрайера; при этом ограничение сверху на скачок ветвления соответствует ограничению снизу на нормирование правой части уравнения. Задача построения расширений с заданной группой Галуа произвольного конечного порядка не решена. В работах Инабы рассматривались p-расширения полей характеристики p, заданные матричным уравнением $$X^{(p)}=AX$$, которое мы здесь называем уравнением Инабы. В этом уравнении $$X^{(p)}$$ обозначает матрицу, полученную возведением каждого элемента квадратной матрицы X в степень p, а - некоторая унипотентная матрица A над данным полем. Такое уравнение задает последовательность расширений полей, каждое из которых задано уравнением Артина-Шрайера. Было доказано, что любое уравнение Инабы задает расширение Галуа, и обратно, любое конечное p-расширение Галуа задается уравнением такого вида. В настоящей работе для полей смешанной характеристики доказано, что расширение, задаваемое уравнением Инабы, является расширением Галуа, если нормирования элементов матрицы удовлетворяют некоторым оценкам снизу, т.е. если скачки промежуточных расширений степени p достаточно малы. Данная конструкция может применяться при решении задачи погружения расширений полей. Уравнение Инабы задает последовательность расширений полей, полученную последовательным присоединением элементов диагоналей матрицы. Это означает, что, если расширение L/K задано уравнением Инабы, и матрица A выбрана так, что на диагоналях с большими номерами записаны нули, то можно получать расширения, содержащие L/K, заменяя нули другими элементами. В работе доказано, что любое нециклическое расширение степени $$p^2$$ с достаточно маленькими скачками можно погрузить в расширение с группой Галуа, изоморфнной группе унипотентных матриц $$3\times 3$$ над полем из p элементов. В конце статьи сформулирован ряд открытых вопросов, при исследовании которых, возможно, окажется полезной данная конструкция.

Relations in the maximal pro-𝑝 quotients of absolute Galois groups

We observe that some fundamental constructions in Galois theory can be used to obtain interesting restrictions on the structure of Galois groups of maximal p p -extensions of fields containing a primitive p p -th root of unity. This is an extension of some significant ideas of Demushkin, Labute, and Serre from local fields to all fields containing a primitive p p -th root of unity. Our techniques use certain natural simple Galois extensions together with some considerations in Galois cohomology and Massey products.

From Periods to Anabelian Geometry and Quantum Amplitudes

Periods are algebraic integrals, extending the class of algebraic numbers, and playing a central, dual role in modern Mathematical-Physics: scattering amplitudes and coefficients of de Rham isomorphism. The Theory of Periods in Mathematics, with their appearance as scattering amplitudes in Physics, is discussed in connection with the Theory of Motives, which in turn is related to Conformal Field Theory (CFT) and Topological Quantum Field Theory (TQFT), on the physics side. There are three main contributions. First, building a bridge between the Theory of Algebraic Numbers and Theory of Periods, will help guide the developments of the later. This suggests a relation between the Betti-de Rham theory of periods and Grothendieck’s Anabelian Geometry, towards perhaps an algebraic analog of Hurwitz Theorem, relating the algebraic de Rham cohomology and algebraic fundamental group, both pioneered by A. Grothendieck. Second, a homotopy-homology refinement of the Theory of Periods will help explain the connections with quantum amplitudes. The novel approach of Yves Andre to Motives via representations of categories of diagrams, relates from a physical point of view to generalized TQFTs. Finally, the known “universality” of Galois Theory, as how symmetries “grow”, controlling the structure of the objects of study, is discussed, in relation to the above several areas of research, together with ensuing further insight into the Mathematical-Physics symbiosis. To better understand and investigate Kontsevich-Zagier conjecture on abstract periods, the article ponders on the case of algebraic Riemann Surfaces representable by Belyi maps. Reformulation of cohomology of cyclic groups as a discrete analog of de Rham cohomology and the Arithmetic Galois Theory will provide a purely algebraic toy-model of the said algebraic homology/homotopy group theory of Grothendieck as part of Anabelian Geometry. The corresponding Platonic Trinity 5,7,11/TOI/E678 leads to connections with ADE-correspondence, and beyond, e.g. Theory of Everything (TOE) and ADEX-Theory. In perspective of the “Ultimate Physics Theory”, quantizing “everything”, i.e. cyclotomic quantum phase and finite Platonic-Hurwitz geometry of qubit frames as baryons, could perhaps be “The Eightfold (Petrie polygon) Way” to finally understand what quark flavors and fermion generations really are.

Dynamical and Algebraic Analysis of Planar Polynomial Vector Fields Linked to Orthogonal Polynomials

In the present work, our goal is to establish a study of some families of quadratic polynomial vector fields connected to orthogonal polynomials that relate, via two different points of view, the qualitative and the algebraic ones. We extend those results that contain some details related to differential Galois theory as well as the inclusion of Darboux theory of integrability and the qualitative theory of dynamical systems. We conclude this study with the construction of differential Galois groups, the calculation of Darboux first integral, and the construction of the global phase portraits.

N-body Dynamics on an Infinite Cylinder: the Topological Signature in the Dynamics

The formulation of the dynamics of N-bodies on the surface of an infinite cylinder is considered. We have chosen such a surface to be able to study the impact of the surface’s topology in the particle’s dynamics. For this purpose we need to make a choice of how to generalize the notion of gravitational potential on a general manifold. Following Boatto, Dritschel and Schaefer [5], we define a gravitational potential as an attractive central force which obeys Maxwell’s like formulas. As a result of our theoretical differential Galois theory and numerical study — Poincare sections, we prove that the two-body dynamics is not integrable. Moreover, for very low energies, when the bodies are restricted to a small region, the topological signature of the cylinder is still present in the dynamics. A perturbative expansion is derived for the force between the two bodies. Such a force can be viewed as the planar limit plus the topological perturbation. Finally, a polygonal configuration of identical masses (identical charges or identical vortices) is proved to be an unstable relative equilibrium for all N > 2.

Integrability of stochastic birth-death processes via differential Galois theory

Stochastic birth-death processes are described as continuous-time Markov processes in models of population dynamics. A system of infinite, coupled ordinary differential equations (the so-called master equation) describes the time-dependence of the probability of each system state. Using a generating function, the master equation can be transformed into a partial differential equation. In this contribution we analyze the integrability of two types of stochastic birth-death processes (with polynomial birth and death rates) using standard differential Galois theory. We discuss the integrability of the PDE via a Laplace transform acting over the temporal variable. We show that the PDE is not integrable except for the case in which rates are linear functions of the number of individuals.

Algebraic independence of generic Painlevé transcendents: PIII and PVI

We prove that if y ′ ′ = f ( y , y ′ , t , α , β , γ , δ ) is a generic Painlevé equation from among the classes I I I and V I , and if y 1 , … , y n are distinct solutions, then t r · d e g C ( t ) C ( t ) ( y 1 , y 1 ′ , … , y n , y n ′ ) = 2 n , that is y 1 , y 1 ′ , … , y n , y n ′ are algebraically independent over C ( t ) . This improves the results obtained by the author and Pillay and completely proves the algebraic independence conjecture for the generic Painlevé transcendents. In the process, we also prove that any three distinct solutions of a Riccati equation are algebraic independent over C ( t ) , provided that there are no solutions in the algebraic closure of C ( t ) . This uses techniques and results from differential Galois theory and answers a very natural question in the theory of Riccati equations.

Strict monadic topology I: First separation axioms and reflections

Given a monad T on the category of sets, we consider reflections of Alg(T) into its full subcategories formed by algebras satisfying natural counterparts of topological separation axioms T0, T1, T2, Tts, and Tths; here ts stands for totally separated and ths for what we call totally homomorphically separated, which coincides with ts in the (compact Hausdorff) topological case. We ask whether these reflections satisfy simple conditions useful in categorical Galois theory, and give some partial answers in easy cases.

Dynamics and integrability analysis of two pendulums coupled by a spring

We consider a system of two simple pendulums, which interact by elastic forces according to Hooke’s law in the presence of the gravitational field. Analysis of global dynamics of the system by means of the Poincaré sections, the bifurcation diagrams and Lyapunov’s exponents, suggest its non-integrability. We give an analytical proof of this fact. An analysis through differential Galois theory is done, providing non-integrability of the system for most of the values of the parameters. For a certain co-dimension one in parameter space family we did not obtain integrability obstructions due to solvability of the first-order variational equations of the Lamé type. We show an application of the higher-order variational equations for proving the non-integrability in this case. As the final result, we received that the system of two pendulums coupled by a spring is integrable only in two cases postulated in the main theorem of the work.