Galois Theory and Spectral theory. Preliminary report.

The aim of this talk is to show an application of Differential Galois Theory in Spectral Theory. In a particular case, we analyze the integrability and the Galois groups of the stationary Schroedinger equation. For example, if the potential is a polynomial, then the Galois group of the Schroedinger equation is a connected non abelian group. On the other hand, if the potential is not a rational function, but there exists a hamiltonian change of variable, then we can algebrize the differential equation preserving the identity component of the Galois group in the original Schroedinger equation, this is the case of Lame equation and Mathieu equation. Finally, we can generate families of Schroedinger equations using the Darboux transformation, Kovacic algorithm and operators theory, where the principal fact is that the Darboux transformation is covariant, isogaloissian and isospectral transformation. This facts play an important role in quantum mechanics.

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Symplectic Properties of the Space of Differential Equations in the Space of Logarithmic Systems.

Let $n \geq 1$, let $S$ be a finite set of points of the Riemann sphere, and let $\cal M$ be the moduli space of irreducible fuchsian systems of rank $n$ with logarithmic singularities lying in $S$ and given ``generic" local monodromies. This space is naturally endowed with a symplectic structure $\omega$. Let further $\cal E$ be the space of irreducible fuchsian differential equations of order $n$, with singularities lying in $S$ and same local monodromies. Following a construction of van der Put and Singer, we can locally embed $\cal E$ as a subspace $\cal N$ of $\cal M$. As remarked by N. Katz, the dimension of $\cal N$ is half the dimension of $\cal M$. We elaborate on this remark by proving that $\cal N$ is a lagrangian subspace of $\cal M$ relatively to the symplectic structure $\omega$.

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Arithmetic Partial Differential Equations

We develop an arithmetic analogue of linear partial differential equations in two independent ``space-time'' variables. The spatial derivative is a Fermat quotient operator, while the time derivative is a usual derivation. This allows us to ``flow'' integers or, more generally, points on algebraic groups with coordinates in rings with arithmetic flavor. In particular, we show that elliptic curves have certain canonical ``arithmetic flows'' on them that are arithmetic analogues of the convection, heat, and wave equations. The same is true for the additive and the multiplicative group.

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Borel-Laplace summation of q-series and confluence. Preliminary report.

We will explain the issues of confluence for Borel-Laplace summation through some examples. Then we will give some partial answer to the general problem. This is a joint work in progress with Changgui Zhang.

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Local differential galois group and adjoint representation.

We construct a (generally) maximal torus containing the exponential torus and develop an algorithm to reduce the weight subspaces of dimension higher than $1$ to root subspaces. We also study the regularity of the exponential torus in the local differential Galois group.

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Canonical representation of radical differential ideals

For every radical differential ideal, one can compute a decomposition into prime (or characterizable) components, which allows to test ideal membership. This representation of the radical differential ideal is not unique in three respects:

1. The components are not unique.
2. The representation of each component by a characteristic set is not unique.
3. The decomposition and representation of each component depend on the choice of ranking on derivatives.

We will discuss how to make the representation unique, namely:

1. A prime decomposition uniquely determined by the radical differential ideal can be computed by extending the algorithm for testing inclusion of quasi-algebraic sets proposed by W. Sit.
2. The canonical characteristic set of a prime differential ideal can be obtained by imposing restrictions proposed by F. Boulier et al. We list some of its properties.
3. In particular, the canonical characteristic set defines a differential analogue of the Gröbner cone. This will lead us to an algorithm that computes a ranking-independent universal characteristic decomposition of a radical differential ideal.

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Iterative q-difference Galois Theory. Preliminary report.

For the beginning, the Galois theory of $q$-difference equations has been built for $q$ non equal to a root of unity. This choice was made for not increasing the field of constants to a transcendental field. However, Peter Hendricks, has studied this problem for $q^m=1$ in his PhD work under the supervision of Marius van der Put. Bu he built a fiber functor from the category of $q$-difference modules over $\mathbb{C}(z)$ with value in the category $Vect_{\mathbb{C}(z^m)}$ of vector spaces of finite dimension over $\mathbb{C}(z^m)$. But this construction is not totally satisfying and to stay in the spirit of Kolchin, we do not want to have such transcendental base fields for Galois groups.

For $q$-difference theory, the problem is not the characteristic but the roots of unity. Inspired by the work of B.H. Matzat and Marius van der Put for Differential Galois theory in positive characteristic, we consider also a family of iterative difference operator instead of considering, just one difference operator, and by this way we stop the increasing of the constant field and succeed to set up a Picard-Vessiot Theory for $q$-difference equations where $q$ is a root of unity and relate it to a Tannakian approach.

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Patching and differential Galois groups. Preliminary report.

Patching methods (building a global object by building it locally) are an important tool for solving inverse problems in classical Galois theory. In this talk, we describe a new formulation of patching over fields, which can be used to patch differential modules. We explain applications to the realization of differential Galois groups.

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Differential Central Simple Algebras and Non-commutative Picard-Vessiot Cocycles

Let $K$ be a differential field of characteristic zero with algebraically closed subfield of constants $C$. A differential central simple algebra, and in particular a differential matrix algebra, over $K$ is trivialized by a Picard-Vessiot extension $E$ of $K$. This yields a bijection between isomorphism classes of differential algebras and Picard-Vessiot cocycles $Z^1(G(E/K),PGL_n(C))$ which cobound in $Z^1(G(E/K),PGL_n(E))$. We will prove these results and illustrate how the differential Brauer group of an algebraically closed field can be non trivial.

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PGL₃ as a differential Galois group

A Picard-Vessiot extension $M/K$ with differential Galois group $G$ is the function field of a $G$-torsor. The $G$-torsors are classified by the non-Abelian cohomology $H^1(K,G)$. In cases where this cohomology can be suitably "parametrised", this allows us to describe the structure of the Picard-Vessiot extensions. This approach will be illustrated in the case of the projective linear group $\mathrm{PGL}_3$.

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Dimension of Difference Field Extensions

In this talk we consider properties of main dimensional characteristics of a finitely generated difference field extension: difference dimension polynomials and the limit degree of the extension. In particular, we show that if the difference transcendental degree of a finitely generated difference field extension G/F is zero, then G contains a subfield H such that the extension G/H is algebraic and H is a finitely generated difference field extension of F with respect to a smaller set of basic translations. We also discuss the relation of the limit degree to the problem of compatibility of difference field extensions.

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Subfields of the complete Picard-Vessiot closure of a differential field. Preliminary report.

The Picard--Vessiot closure $(E)_1$ of a differential field $E$ (differential fields always assumed to have algebraically closed characteristic zero field of constants) is the compositum of all its Picard--Vessiot extensions. If $F$ is a differential field, its complete Picard--Vessiot closure $F_\infty$ is $\cup_{i \geq 0} F_i$ where $F_0=F$ and $F_{i+1}=(F_i)_1$. There is a semi--Galois correspondence between all differential subfields of $F_\infty$ over $F$ and subgroups of the group $G$ of all differential automorphisms of $F_\infty$ over $F$. We characterize the (differentially) finitely generated subfields of $F_\infty$ (containing $F$).

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Differential Equations and Frobenius Structures

The (strong) Frobenius structure for differential equations was introduced by B. Dwork for $p$-adic differential equations. In the case of positive characteristic the existence of a (strong) Frobenius structure is equivalent to the finiteness of the differential Galois group. This fact can be used, for example, to construct additive polynomials with given Galois group and to develop algebraicity criteria in characteristic zero.

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A generalization of the Riemann-Hilbert Problem.

We discuss the existence of systems of linear ordinary differential equations with coefficients in ${\mathbb C}(z)$ that satisfy generalized monodromy data at prescribed, possibly irregular, singularities. This inverse problem reduces to the classical Riemann-Hilbert problem if all the singularities are required to be Fuchsian, and to the Birkhoff standard form problem if there are exactly two, one of which Fuchsian, prescribed singularities. This generalized Riemann-Hilbert problem is naturally related to the inverse problem in differential Galois theory over ${\mathbb C}(z)$ as far as one is concerned with the Poincaré rank of the singularities. The talk presents joint work with the late Andrey A. Bolibrukh and Stéphane Malek.

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Jacobi's work on normal forms of differential systems. Preliminary report.

In 1866 was first published Jacobi's posthumous paper "The reduction to normal form of a non-normal system of differential equations" (in latin). A method is given there to compute a normal form of a system $P_i=0$, using a minimal number $\ell_{i}$ of derivatives of $P_i$. The given bound is generically true and sharp. The $\ell_{i}$ may be computed using the algorithm Jacobi gave to compute ``Jacobi's bound'' on the system order, a forgotten ancestor of Kuhn's Hungarian method for the assignment problem (1955).

He also provides a generic method to eliminate all variables except one, using again as few derivatives as possible, a very interesting result for improving the algorithmic complexity of a resolvent computation.

These are described using the formalism of differential algebra in order to give precise proofs following Jacobi's ideas. The content of some unpublished part of manuscript II/13b (Jacobis Nachlaß, Archiv der Berlin-Brandenburgischen Akademie der Wissenschaften) will also be exposed. Jacobi considers there the more general problem of finding all possible normal forms for a given system, giving precise conditions for a system of order $n$ in two variables to have less than $n+1$ possible normal forms (or caracteristic sets) for all orderings.

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Tannakian formalism for linear differential algebraic groups.

Tannaka's Theorem states that a linear algebraic group $G$ is determined by the category of finite dimensional $G$-modules and the forgetful functor. We extend this result to linear differential algebraic groups by introducing a category corresponding to their representations and discuss how this category determines such a group.\par

We also provide conditions for a category with a fiber functor to be equivalent to the category of representations of a linear differential algebraic group. This generalizes the notion of a neutral Tannakian category used to characterize the category of representations of a linear algebraic group.

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Standard bases in differential algebra.

Constructive methods in rings of differential polynomials are connected, first of all, with characteristic sets of differential ideals whose theory was developed by Ritt and Kolchin for prime differential ideals. Later it had been extended to a larger class of perfect differential ideals.

In the case of linear partial differential polynomials, this theory may be treated as the theory of Gr\"obner bases of differential modules, to which all methods and approaches of commutative Gröbner bases are applicable, in particular, the theory of staggered bases by Gebauer and Möller. Specifying some parameters in their algorithm, we obtain Janet's bases, a particular case of involutive bases.

Ollivier and Carra-Ferro proposed a definition of standard bases of differential ideals based on admissible orderings of differential monomials. Unfortunately, this basis is infinite for most of differential ideals (e.g., the ideal $[y^2]$). Investigations in this area had been suspended for a long time.\par

Zobnin discovered that these bases become finite if, instead of the lexicographic ordering, we consider other orderings of differential monomials. This fact revived the interest in this subject and initiated the study of orderings of differential monomials.

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A theorem of Sit.

In 1975, Sit showed that the set of Kolchin (dimension) polynomials is well ordered by eventual dominance. We will give an order-theoretic proof of this theorem and consider its applications in the model theory of differential fields.

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O-minimality and quantifier elimination in some non quasi-analytic classes. Preliminary report.

\def \R {\mathbb{R}}

I extend the results of $[R1]$ which deal with classes of restricted real quasi-analytic functions, to classes of non quasi-analytic functions.

More precisely, in $[R1]$ only classes of functions, $C^{\infty}$ on a whole compact box of $\R^n$ and quasi-analytic on this box, were considered. Now, we study some well-closed classes of functions, $C^{\infty}$ on an open bounded box, continuous on the closure of this box and which satisfy a condition of non-degeneration (equivalent to quasi-analycity in the former case), expressed via model theory. For example, certain of these classes come from solutions of differential equations.

I obtain, like in $[R1]$, results of o-minimality (which generalize for example those of $[vdDS]$) and of quantifier elimination, which imply in particular, preparation theorems in the considered classes.

$[vdDS]$: L. van den Dries and P. Speissegger, "The real field with convergent generalized power series", Trans. Amer. Math. Soc., 350 (1998), 4377-4421.

$[R1]$: A. Rambaud, "Quasi-analycité, o-minimalité et élimination des quantificateurs", Ph.D. thesis, Université Paris 7, 2005.

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Spectra of rings differentially finitely generated over a subring.

We consider differential rings that are algebras over the field of rational numbers. We present some ideas that are useful for investigation of such rings.

One of them uses the relation between the spectrum of an arbitrary differential ring and its differential spectrum. We consider pairs of properties. One of them characterizes the spectrum and the other one does it for the differential spectrum. If the first property holds, the other one is satisfied as well. The described pairs of properties allow us to reduce the study of a differential ring to the study of this ring considered as an ordinary ring.

To prove a theorem describing the structure of differential integral domains differentially finitely generated over a subring, we apply results about characteristic sets of differential ideals of the ring of differential polynomials over an integral domain.

The main proved theorems enable us to reduce the proof of propositions in differential algebra to the proof of some propositions of commutative algebra. We distinguish a very dense subset of spectrum with good properties and discuss analogues of differential algebraic varieties. Also, as an illustration of the presented method, some analogues of geometric theorems are proved without using the universal field.

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Analytic q-difference equations, universal rings and universal Galois groups.

$q$ is a complex number satisfying $0<|q|<1$ and $K={\bf C}(\{z\})$ is the field of the convergent Laurent series. The automorphism $\phi$ of $K$ given by $\phi (z)=q z$ makes $K$ into a difference field. A difference module is a finite dimensional vector space over $K$, provided with a bijective map $\Phi$ satisfying $\Phi (f\cdot m)=\phi (f)\cdot \Phi (m)$. A difference module has a Picard-Vessiot ring and a (difference) Galois group. \par One also considers difference modules over the difference field of the formal Laurent series $\widehat{K}={\bf C}((z))$. For the latter category of modules we will give an explicit description of the universal difference ring and its universal Galois group. For the category of the difference modules over $K$ we present a tentative description of the universal difference ring and its corresponding universal Galois group.

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Gröbner bases in difference-differential modules and their applications.

Recently we have introduced a construction of Gröbner bases for difference-differential (d-d) modules, based on a new concept of generalized term ordering for exponent vectors over the integers. We further investigate the key concept of S-polynomial for such difference-differential bases. We also apply the method to compute the difference-differential dimension polynomial of a d-d module and of a system of linear partial difference-differential equations. This is joint work with M. Zhou of Beihang University in Beijing.

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Symbolic-numeric Computation of Implicit Riquier Bases for PDE.

Riquier Bases for systems of analytic PDE are, loosely speaking, a differential analogue of Grobner Bases for polynomial equations. They are determined in the exact case by applying a sequence of prolongations and eliminations to an input system of PDE.

We present a symbolic-numeric method to determine Riquier Bases in implicit form for systems which are dominated by pure derivatives in one of the independent variables and have the same number of PDE and unknowns.

The method is successful provided the prolongations with respect to the dominant independent variable have a block structure which is uncovered by Linear Programming and certain Jacobians are non-singular when evaluated at points on the zero sets defined by the functions of the PDE. For polynomially nonlinear PDE, homotopy continuation methods from Numerical Algebraic Geometry can be used to compute approximations of the points.

We give a differential algebraic interpretation of Pryce's method for ODE, which generalizes to the PDE case. A major aspect of the method's efficiency is that only prolongations with respect to a single (dominant) independent variable are made, possibly after a random change of coordinates.

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Factorization in Skew Polynomial Rings.

Efficient algorithms are presented for factoring polynomials in the skew polynomials over complex number field and quantum planes.

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Created by the KSDA Organizing Committee Last updated August 13, 2007