Difference Galois Group Research Articles

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.

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.

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.

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 $\operatorname{GL}_{2}(\mathbb C)$.

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.

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.

An Algorithm Determining the Difference Galois Group of Second Order Linear Difference Equations

An Algorithm Determining the Difference Galois Group of Second Order Linear Difference Equations

Algebraic integrability of the two-body Ruijsenaars operator

We study the algebra of difference operators that commute with the two-body Ruijsenaars operator, a $q$-deformation of the Lam\'e differential operator, for generic values of the deformation parameter. The algebra is commutative. It is the algebra of polynomial functions on an affine hyperelliptic curve $Y^2=P(X^2)$. We also compute the difference Galois group of the eigenvalue problem.

Algebraic integrability of the two-body Ruijsenaars operator

We study the algebra of difference operators that commute with the two-body Ruijsenaars operator, a $q$-deformation of the Lam\'e differential operator, for generic values of the deformation parameter. The algebra is commutative. It is the algebra of polynomial functions on an affine hyperelliptic curve $Y^2=P(X^2)$. We also compute the difference Galois group of the eigenvalue problem.