Picard Vessiot Research Articles

Spectral Curves for Third-Order ODOs

Spectral curves are algebraic curves associated to commutative subalgebras of rings of ordinary differential operators (ODOs). Their origin is linked to the Korteweg–de Vries equation and to seminal works on commuting ODOs by I. Schur and Burchnall and Chaundy. They allow the solvability of the spectral problem Ly=λy, for an algebraic parameter λ and an algebro-geometric ODO L, whose centralizer is known to be the affine ring of an abstract spectral curve Γ. In this work, we use differential resultants to effectively compute the defining ideal of the spectral curve Γ, defined by the centralizer of a third-order differential operator L, with coefficients in an arbitrary differential field of zero characteristic. For this purpose, defining ideals of planar spectral curves associated to commuting pairs are described as radicals of differential elimination ideals. In general, Γ is a non-planar space curve and we provide the first explicit example. As a consequence, the computation of a first-order right factor of L−λ becomes explicit over a new coefficient field containing Γ. Our results establish a new framework appropriate to develop a Picard–Vessiot theory for spectral problems.

Burchnall–Chaundy polynomials for matrix ODOs and Picard–Vessiot Theory

Burchnall and Chaundy showed that if two ordinary differential operators (ODOs) P, Q with analytic coefficients commute then there exists a polynomial f(λ,μ) with complex coefficients such that f(P,Q)=0, called the BC-polynomial. This polynomial can be computed using the differential resultant for ODOs. In this work we extend this result to matrix ordinary differential operators, MODOs. Our matrices have entries in a differential field K, whose field of constants C is algebraically closed and of zero characteristic. We restrict to the case of order one operators P, with invertible leading coefficient. We define a new differential elimination tool, the matrix differential resultant. We use it to compute the BC-polynomial f of a pair of commuting MODOs, and we also prove that it has constant coefficients. This resultant provides the necessary and sufficient condition for the spectral problem PY=λY,QY=μY to have a solution. Techniques from differential algebra and Picard–Vessiot theory allow us to describe explicitly isomorphisms between commutative rings of MODOs C[P,Q] and a finite product of rings of irreducible algebraic curves.

Spectral Picard–Vessiot fields for Algebro-geometric Schrödinger operators

This work is a galoisian study of the spectral problem $L\Psi=\lambda\Psi$, for algebro-geometric second order differential operators $L$, with coefficients in a differential field, whose of constants $C$ is algebraically closed and of characteristic zero. Our approach regards the spectral parameter $\lambda$ an algebraic variable over $C$, forcing the consideration of a new of coefficients for $L-\lambda$, whose of constants is the $C(\Gamma)$ of the spectral curve $\Gamma$. Since $C(\Gamma)$ is no longer algebraically closed, the need arises of a new algebraic structure, generated by the solutions of the spectral problem over $\Gamma$, called Spectral Picard-Vessiot field of $L-\lambda$. An existence theorem is proved using differential algebra, allowing to recover classical Picard-Vessiot theory for each $ \lambda = \lambda_0 $. For rational spectral curves, the appropriate algebraic setting is established to solve $L\Psi=\lambda\Psi$ analitically and to use symbolic integration. We illustrate our results for Rosen-Morse solitons.

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.

Differential Galois cohomology and parameterized Picard–Vessiot extensions

Assuming that the differential field [Formula: see text] is differentially large, in the sense of [León Sánchez and Tressl, Differentially large fields, preprint (2020); arXiv:2005.00888], and “bounded” as a field, we prove that for any linear differential algebraic group [Formula: see text] over [Formula: see text], the differential Galois (or constrained) cohomology set [Formula: see text] is finite. This applies, among other things, to closed ordered differential fields in the sense of [Singer, The model theory of ordered differential fields, J. Symb. Logic 43(1) (1978) 82–91], and to closed[Formula: see text]-adic differential fields in the sense of [Tressl, The uniform companion for large differential fields of characteristic [Formula: see text], Trans. Amer. Math. Soc. 357(10) (2005) 3933–3951]. As an application, we prove a general existence result for parameterized Picard–Vessiot (PPV) extensions within certain families of fields; if [Formula: see text] is a field with two commuting derivations, and [Formula: see text] is a parameterized linear differential equation over [Formula: see text], and [Formula: see text] is “differentially large” and [Formula: see text] is bounded, and [Formula: see text] is existentially closed in [Formula: see text], then there is a PPV extension [Formula: see text] of [Formula: see text] for the equation such that [Formula: see text] is existentially closed in [Formula: see text]. For instance, it follows that if the [Formula: see text]-constants of a formally real differential field [Formula: see text] is a closed ordered[Formula: see text]-field, then for any homogeneous linear [Formula: see text]-equation over [Formula: see text] there exists a PPV extension that is formally real. Similar observations apply to [Formula: see text]-adic fields.

Admissible Galois Structures on the categories dual to some varieties of universal algebras

Abstract The subject of the paper is suggested by G. Janelidze and motivated by his earlier result giving a positive answer to the question posed by S. MacLane whether the Galois theory of homogeneous linear ordinary differential equations over a differential field (which is Kolchin–Ritt theory and an algebraic version of Picard–Vessiot theory) can be obtained as a particular case of G. Janelidze’s Galois theory in categories. One ground category in the Galois structure involved in this theory is dual to the category of commutative rings with unit, and another one is dual to the category of commutative differential rings with unit. In the present paper, we apply the general categorical construction, the particular case of which gives this Galois structure, by replacing “commutative rings with unit” by algebras from any variety V \mathscr{V} of universal algebras satisfying the amalgamation property and a certain condition (of the syntactical nature) for elements of amalgamated free products which was introduced earlier, and replacing “commutative differential rings with unit” by V \mathscr{V} -algebras equipped with additional unary operations which satisfy some special identities to construct a new Galois structure. It is proved that this Galois structure is admissible. Moreover, normal extensions with respect to it are characterized in the case where V \mathscr{V} is any of the following varieties: abelian groups, loops and quasigroups.

Generalized Picard–Vessiot extensions and differential Galois cohomology

In an earlier paper it was proved that if a differential field $(K,\delta)$ is algebraically closed and closed under Picard-Vessiot extensions then every differential algebraic principal homogeneous space over K for a linear differential algebraic group G over K has a K-rational point (in fact if and only if). This paper explores whether and if so, how, this can be extended to (a) several derivations, (b) one automorphism. Under a natural notion of generalized Picard-Vessiot extension (in the case of several derivations), we give a counterexample. We also have a counterexample in the case of one automorphism. We also formulate and prove some positive statements in the case of several derivations.

Picard–Vessiot groups of Lauricella's hypergeometric systems EC and Calabi–Yau varieties arising integral representations

We study the Zariski closure of the monodromy group Mon of Lauricella's hypergeometric function F C . If the identity component Mon 0 acts irreducibly, then Mon ¯ ∩ SL 2 n ( C ) must be one of classical groups SL 2 n ( C ) , SO 2 n ( C ) and Sp 2 n ( C ) . We also study Calabi–Yau varieties arising from integral representations of F C .

Matrices of Scalar Differential Operators: Divisibility and Spaces of Solutions

We investigate the connection between divisibility of full-rank square matrices of linear scalar differential operators over some differential field K, and the solution spaces of these matrices over the universal Picard–Vessiot extension of K. We establish some properties of the solution spaces of the greatest common right divisor and the least common left multiple of such matrices.

Triviality of differential Galois cohomology of linear differential algebraic groups

For a partial differential field K, we show that the triviality of the first differential Galois cohomology of every linear differential algebraic group over K is equivalent to K being algebraically, Picard–Vessiot, and linearly differentially closed. This cohomological triviality condition is also known to be equivalent to the uniqueness up to an isomorphism of a Picard–Vessiot extension of a linear differential equation with parameters.

Solvability of Equations by Quadratures and Newton’s Theorem

Picard–Vessiot theorem (1910) provides a necessary and sufficient condition for solvability of linear differential equations of order n by quadratures in terms of its Galois group. It is based on the differential Galois theory and is rather involved. Liouville in 1839 found an elementary criterium for such solvability for $$n=2$$ . Ritt simplified Liouville’s theorem (1948). In 1973 Rosenlicht proved a similar criterium for arbitrary n. Rosenlicht work relies on the valuation theory and is not elementary. In these notes we show that the elementary Liouville–Ritt method based on developing solutions in Puiseux series as functions of a parameter works smoothly for arbitrary n and proves the same criterium.

The Dynamical Mordell–Lang Conjecture for Skew-Linear Self-Maps. Appendix by Michael Wibmer

Abstract Let $k$ be an algebraically closed field of characteristic $0$, let $N\in{\mathbb{N}}$, let $g:{\mathbb{P}}^1{\longrightarrow } {\mathbb{P}}^1$ be a nonconstant morphism, and let $A:{\mathbb{A}}^N{\longrightarrow } {\mathbb{A}}^N$ be a linear transformation defined over $k({\mathbb{P}}^1)$, that is, for a Zariski-open dense subset $U\subset{\mathbb{P}}^1$, we have that for $x\in U(k)$, the specialization $A(x)$ is an $N$-by-$N$ matrix with entries in $k$. We let $f:{\mathbb{P}}^1\times{\mathbb{A}}^N{\dashrightarrow } {\mathbb{P}}^1\times{\mathbb{A}}^N$ be the rational endomorphism given by $(x,y)\mapsto (\,g(x), A(x)y)$. We prove that if $g$ induces an automorphism of ${\mathbb{A}}^1\subset{\mathbb{P}}^1$, then each irreducible curve $C\subset{\mathbb{A}}^1\times{\mathbb{A}}^N$ that intersects some orbit $\mathcal{O}_f(z)$ in infinitely many points must be periodic under the action of $f$. Furthermore, in the case $g:{\mathbb{P}}^1{\longrightarrow } {\mathbb{P}}^1$ is an endomorphism of degree greater than $1$, then we prove that each irreducible subvariety $Y\subset{\mathbb{P}}^1\times{\mathbb{A}}^N$ intersecting an orbit $\mathcal{O}_f(z)$ in a Zariski dense set of points must be periodic. Our results provide the desired conclusion in the Dynamical Mordell–Lang Conjecture in a couple new instances. Moreover, our results have interesting consequences toward a conjecture of Rubel and toward a generalized Skolem–Mahler–Lech problem proposed by Wibmer in the context of difference equations. In the appendix it is shown that the results can also be used to construct Picard–Vessiot extensions in the ring of sequences.

Schwarz maps of algebraic linear ordinary differential equations

A linear ordinary differential equation is called algebraic if all its solution are algebraic over its field of definition. In this paper we solve the problem of finding closed form solution to algebraic linear ordinary differential equations in terms of standard equations. Furthermore, we obtain a method to compute all algebraic linear ordinary differential equations with rational coefficients by studying their associated Schwarz map through the Picard–Vessiot Theory.

The Picard–Vessiot theory, constrained cohomology, and linear differential algebraic groups

We prove that a differential field K is algebraically closed and Picard–Vessiot closed if and only if the differential Galois cohomology group, H∂1(K,G), is trivial for any linear differential algebraic group G over K. We give applications to the parameterized Picard–Vessiot theory and pose several problems.

Selected and satellite unknowns in linear differential systems

Consider the differential field K=Q‾(x) with derivation d/dx. Let some unknowns (components of the unknown vector y=(y1,…,yn)T) of a linear homogeneous differential system S over K be selected. Denote this set of selected unknowns by s. An unselected unknown yj of the system S is called satellite for s if the minimal subfield of a Picard–Vessiot extension over K for S, that contains all selected components of all solutions to S, also contains yj component of any solution. We present an algorithm for constructing the set of satellite unknowns for a given linear homogeneous differential system with selected unknowns.

Generic Picard–Vessiot Extensions for Nonconnected Groups

Let K be a differential field with algebraically closed field of constants 𝒞 and G a linear algebraic group over 𝒞. We provide a characterization of the K-irreducible G-torsors for nonconnected groups G in terms of the first Galois cohomology H1(K, G) and use it to construct Picard–Vessiot extensions which correspond to nontrivial torsors for the infinite quaternion group, the infinite multiplicative and additive dihedral groups and the orthogonal groups. The extensions so constructed are generic for those groups.

Some applications of parameterized Picard–Vessiot theory

This is an expository article describing some applications of parameterized Picard–Vessiot theory. This Galois theory for parameterized linear differential equations was Cassidy and Singer's contribution to an earlier volume dedicated to the memory of Andrey Bolibrukh. The main results we present here were obtained for families of ordinary differential equations with parameterized regular singularities in joint work with Singer. They include parametric versions of Schlesinger's theorem and of the weak Riemann–Hilbert problem as well as an algebraic characterization of a special type of monodromy evolving deformations illustrated by the classical Darboux–Halphen equation. Some of these results have recently been applied by different authors to solve the inverse problem of parameterized Picard–Vessiot theory, and were also generalized to irregular singularities. We sketch some of these results by other authors. The paper includes a brief history of the Darboux–Halphen equation as well as an appendix on differentially closed fields.

On the computation of the parameterized differential Galois group for a second-order linear differential equation 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∂2∂x2Y+r1∂∂xY+r0Y=0, where the coefficients r1 and r0 belong to the field of rational functions F(x) over a computable Π-field F of characteristic zero, and the finite set of commuting derivations Π is thought of as consisting of derivations with respect to parameters. This work relies on earlier procedures developed by Dreyfus and by the present author to compute G under the assumption that r1=0, which guarantees that G is unimodular. When r1≠0, 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 Π-transcendental over the underlying Π-field of F(x). We give various examples of computation and some applications to differential transcendence.

Real and p-adic Picard–Vessiot fields

We consider differential modules over real and p-adic differential fields K such that its field of constants k is real closed (resp., p-adically closed). Using P. Deligne’s work on Tannakian categories and a result of J.-P. Serre on Galois cohomology, a purely algebraic proof of the existence and unicity of real (resp., p-adic) Picard–Vessiot fields is obtained. The inverse problem for real forms of a semi-simple group is treated. Some examples illustrate the relations between differential modules, Picard–Vessiot fields and real forms of a linear algebraic group.

Skolem–Mahler–Lech type theorems and Picard–Vessiot theory

We show that three problems involving linear difference equations with rational function coefficients are essentially equivalent. The first problem is the generalization of the classical Skolem–Mahler–Lech theorem to rational function coefficients. The second problem is whether or not for a given linear difference equation there exists a Picard–Vessiot extension inside the ring of sequences. The third problem is a certain special case of the dynamical Mordell–Lang conjecture. This allows us to deduce solutions to the first two problems in a particular but fairly general special case.