Differential Galois Research Articles

Difference sheaves and torsors

We develop sheaf theory in the context of difference algebraic geometry. We introduce categories of difference sheaves and develop the appropriate cohomology theories. As specializations, we get difference Galois cohomology, difference Picard group and a

Difference Galois theory and dynamics

We develop a Galois theory for difference ring extensions, inspired by Magid's separable Galois theory for ring extensions and by Janelidze's categorical Galois theory. Our difference Galois theorem states that the category of difference ring extensions split by a chosen Galois difference ring extension is classified by actions of the associated difference profinite Galois groupoid. In particular, difference locally étale extensions of a difference ring are classified by its difference profinite fundamental groupoid.The emergence of difference profinite spaces, viewed as discrete dynamical systems in the realm of topological dynamics, leads us to investigate the interaction of difference algebra and symbolic dynamics. As an application of this interaction, we prove the near-rationality of a certain difference zeta function counting solutions of systems of difference algebraic equations over algebraic closures of finite fields with Frobenius.

Algebraic groups as difference Galois groups of linear differential equations

We study the inverse problem in the difference Galois theory of linear differential equations over the difference-differential field C ( x ) with derivation d d x and endomorphism f ( x ) ↦ f ( x + 1 ) . Our main result is that every linear algebraic group , considered as a difference algebraic group, occurs as the difference Galois group of some linear differential equation over C ( x ) .

A Density Theorem for the Difference Galois Groups of Regular Singular Mahler Equations

Abstract The difference Galois theory of Mahler equations is an active research area. The present paper aims at developing the analytic aspects of this theory. We first attach a pair of connection matrices to any regular singular Mahler equation. We then show that these connection matrices can be used to produce a Zariski-dense subgroup of the difference Galois group of any regular singular Mahler equation.

On the Nature of Four Models of Symmetric Walks Avoiding a Quadrant

We study the nature of the generating series of some models of walks with small steps in the three quarter plane. More precisely, we restrict ourselves to the situation where the group is infinite, the kernel has genus one, and the step set is diagonally symmetric (i.e., with no steps in anti-diagonal directions). In that situation, after a transformation of the plane, we derive a quadrant-like functional equation. Among the four models of walks, we obtain, using difference Galois theory, that three of them have a differentially transcendental generating series, and one has a differentially algebraic generating series.

Hypertranscendence and linear difference equations

After Hölder proved his classical theorem about the Gamma function, there has been a whole bunch of results showing that solutions to linear difference equations tend to be hypertranscendental (i.e., they cannot be solution to an algebraic differential equation). In this paper, we obtain the first complete results for solutions to general linear difference equations associated with the shift operator x ↦ x + h x\mapsto x+h ( h ∈ C ∗ h\in \mathbb {C}^* ), the q q -difference operator x ↦ q x x\mapsto qx ( q ∈ C ∗ q\in \mathbb {C}^* not a root of unity), and the Mahler operator x ↦ x p x\mapsto x^p ( p ≥ 2 p\geq 2 integer). The only restriction is that we constrain our solutions to be expressed as (possibly ramified) Laurent series in the variable x x with complex coefficients (or in the variable 1 / x 1/x in some special case associated with the shift operator). Our proof is based on the parametrized difference Galois theory initiated by Hardouin and Singer. We also deduce from our main result a general statement about algebraic independence of values of Mahler functions and their derivatives at algebraic points.

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.

Hypertranscendence of solutions of Mahler equations

The last years have seen a growing interest from mathematicians in Mahler functions. This class of functions includes the generating series of the automatic sequences. The present paper is concerned with the following problem, which is rather frequently encountered in combinatorics: a set of Mahler functions u_1, …, u_n being given, are u_1, …, u_n and their successive derivatives algebraically independent? In this paper, we give general criteria ensuring an affirmative answer to this question. We apply our main results to the generating series attached to the so-called Baum–Sweet and Rudin–Shapiro automatic sequences. In particular, we show that these series are hyperalgebraically independent, i.e., these series and their successive derivatives are algebraically independent. Our approach relies on parametrized difference Galois theory (in this context, the algebro-differential relations between the solutions of a given Mahler equation are reflected by a linear differential algebraic group).

Real difference Galois theory

In this paper, we develop a difference Galois theory in the setting of real fields. After proving the existence and uniqueness of the real Picard-Vessiot extension, we define the real difference Galois group and prove a Galois correspondence.

On the algebraic relations between Mahler functions

In the last years, a number of authors have studied the algebraic relations between the generating series of automatic sequences. It turns out that these series are solutions of Mahler type equations. This paper is mainly concerned with the difference Galois groups of Mahler type equations (these groups reflect the algebraic relations between the solutions of the equations). In particular, we study in detail the equations of order 2 2 and compute the difference Galois groups of classical equations related to the Baum-Sweet and to the Rudin-Shapiro automatic sequences.

Picard-Vessiot and categorically normal extensions in differential-difference Galois theory

The connection between categorical and differential Galois theories established by the author (published in 1989) is extended to the context that includes difference Galois theory.

Galois Groups of Difference Equations of Order Two on Elliptic Curves

This paper is concerned with difference equations on elliptic curves. We establish some general properties of the difference Galois groups of equations of order two, and give applications to the calculation of some difference Galois groups. For instance, our results combined with a result from transcendence theory due to Schneider allow us to identify a large class of discrete Lam\'e equations with difference Galois group $\operatorname{GL}_{2}(\mathbb C)$.

Generalized linear cellular automata in groups and difference Galois theory

Generalized non-autonomous linear cellular automata are systems of linear difference equations with many variables. They can be easily transformed into convolution equations in discrete groups. We study these equations from the standpoint of the Galois theory of linear difference equations and discrete Fourier transform.

Lax operators, Poisson groups, and the differential Galois theory

We examine the transfer of a Poisson structure in a space of differential or difference Lax operators to the space of solutions of the corresponding auxiliary problem (the space of wave functions) in detail. We investigate a spontaneous symmetry breaking resulting in the appearance of a nontrivial Poisson structure on the differential or difference Galois group. We review the difference version of the Drinfeld-Sokolov theory and describe a new type of classical r-matrices related to generalized exchange algebras.

Difference Galois theory of linear differential equations

We develop a Galois theory for linear differential equations equipped with the action of an endomorphism. This theory is aimed at studying the difference algebraic relations among the solutions of a linear differential equation. The Galois groups here are linear difference algebraic groups, i.e., matrix groups defined by algebraic difference equations.

A difference version of Nori’s theorem

We consider (Frobenius) difference equations over (F_q(s,t), phi) where phi fixes t and acts on F_q(s) as the Frobenius endomorphism. We prove that every semisimple, simply-connected linear algebraic group G defined over F_q can be realized as a difference Galois group over F_{q^i}(s,t) for some i in N. The proof uses upper and lower bounds on the Galois group scheme of a Frobenius difference equation that are developed in this paper. The result can be seen as a difference analogue of Nori's Theorem which states that G(F_q) occurs as (finite) Galois group over F_q(s).

Poisson Geometry of Difference Lax Operators, and Difference Galois Theory, or Quantum groups from Poisson brackets anomalies

We discuss the lift of Poisson structures associated with auxiliary linear problems for the differential and difference Lax equations to the space of wave functions. Due to a peculiar symmetry breaking, the corresponding differential and difference Galois groups become Poisson Lie Groups.

On a Matrix Representation for Polynomially Recursive Sequences

In this article we derive several consequences of a matricial characterization of P-recursive sequences. This characterization leads to canonical representations of these sequences. We show their uniqueness for a given sequence, up to similarity. We study their properties: operations, closed forms, d'Alembertian sequences, field extensions, positivity, extension of the sequence to $\mathbb Z$, difference Galois group.

Descent for differential Galois theory of difference equations: confluence andq-dependence

The present paper contains two results that generalize and improve constructions of Hardouin and Singer. In the case of a derivation, we prove that the parametrized Galois theory for difference equations constructed by Hardouin and Singer can be descended from a differentially closed to an algebraically closed field. In the second part of the paper, we show that the theory can be applied to deformations of q-series to study the differential dependence with respect to x d d x and q d dq . We show that the parametrized difference Galois group (with respect to a convenient derivation defined in the text) of the Jacobi Theta function can be considered as the Galoisian counterpart of the heat equation.

Hypertranscendance de fonctions de Mahler du premier ordre

Let K be a field equipped with an endomorphism σ. In this Note, we use difference Galois theory to give an algebraic independence criterion for solutions of first order σ-equations. This result allows us to characterize the hyperalgebraic solutions of such σ-equations when K is endowed with a derivation Δ such that Δ ∘ σ = p σ ∘ Δ , where p is a ( σ , Δ ) -constant which is not a root of unity. We deduce from this theorem, in the setting of the Mahler operator, a galoisian proof of a hypertranscendency theorem of Ke. Nishioka.