| About | Participants | Schedule | Location |
Please note: The schedule is also available in PDF format.
| Wednesday (9th Feb.) | Thursday (10th Feb.) | Friday (11th Feb.) | |||||||
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| 8:30–10:00 |
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| 10:00–10:30 | Opening | Coffee break | Coffee break | ||||||
| 10:30–12:00 |
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| 12:00–14:00 | Lunch break | Lunch break | |||||||
| 14:00–15:30 |
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| 15:30–16:00 | Coffee break | Coffee break | |||||||
| 16:00–17:30 |
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On Thursday evening at 7'o clock we will have a joint banquet at the u\hof.
We are working with matrices over a ring K[D,σ,θ] of Ore polynomials over a skew field K. Extending a result of Kojima et al for usual polynomials it is shown that in this setting the Hermite and Popov normal forms correspond to Gröbner bases with respect to certain orders. The FGLM algorithm is adapted to this setting and used for converting Popov forms into Hermite forms and vice versa. The approach works for arbitrary, i.e., not necessarily square matrices where we establish termination criteria to deal with infinitely dimensional factor spaces.
Partial integro-differential operators are created as an algebraic model of Green's operators for linear partial differential equations. Unlike their univariate counterparts, they honor the chain rule and the substitution rule for describing a change of variables (currently restricted to linear transformations). For mastering these rules as well as the univariate ones (Leibniz law and Baxter rule), we propose a new rewrite system that promises to have canonical forms.
(Session chair: F. Winkler)
(Session chair: Z. Li)
Let R be a commutative ring K[x1,…,xn] over a field K of characteristic 0. Moreover, let D=D(R) be the n-th Weyl algebra, that is an associative K-algebra, generated by {x1,…,xn, ∂1,…,∂n} subject to relations ∂jxi=xi∂j+δij ∀ 1≤i,j≤n. Indeed, Weyl algebra is the algebra of linear partial differential operators with polynomial coefficients.
How to compute a (possibly smallest) system of PDE's with polynomial coefficients, such that f∈R∖K is a solution of such system? Since R is finitely presented D(R)-module with the natural action xi•p=xi⋅p, ∂i•p=[∂p/∂xi], we get the answer by computing (using Gröbner bases) a left ideal AnnD(R)f={a∈D(R)|a•f=0}.
We can compute the annihilator of fα for any concrete α∈C as before. D-module theory allows us to compute the annihilator of fs for symbolic s and, moreover, s itself appears in the annihilator AnnD(R)[s]fs⊂D(R)[s]=D(R)⊗K[s] polynomially.
As an application, an algorithm to compute the explicit D(R)-module structure of the localization K[x]F for F={fi|i≥0}⊂R will be demonstrated.
J. Bernstein proved in 1972, that for a polynomial f∈R there exist an operator P(s)∈D(R)[s] and a monic polynomial b(s)∈K[s], such that for any s the equality
holds. bf(s) is called the Bernstein-Sato polynomial of f. It has many interesting properties, which will be discussed in the talk. In particular, bf(s) is quite sensitive with respect to singularities of V(f). For instance, if V(f) is smooth, one can easily show that bf(s)=s+1. Otherwise bf(s) might be very nontrivial and its computation very challenging (cf. the talk by Daniel Andres!). We show, how to compute AnnD(R)[s]fs,bf(s) and Pf(s) effectively.
A very recent development utilizes the central character decomposition for D-modules, arising from fs and leads to an interesting stratification of the hypersurface V(f).
Some important applications of D-modules will be discussed and accompanied by nontrivial live examples, computed with the Singular:Plural's package for D-modules. In particular, we sketch the construction of a generalization of a Bernstein-Sato polynomial and related data to the case of an affine variety V(f1,…,fm).
(Session chair: X.-S. Gao)
In 1918 Emmy Noether showed that there exists a relationship between symmetries and conservation laws in physics. Recently we proved that Noether’s conservation laws could be written as a divergence of the product of a moving frame and a vector of invariants. We will illustrate how the knowledge of the conservation laws structure of SE(3) symmetric variational problems eases the integration problem.
We shall present a survey on some resent results on the algebraic limit cycles of the real planar polynomial fields, and also a list of related open problems.
(Session chair: J.-F. Pommaret)
Let (S) Y'=A(x)Y be a system of first order linear differential equations with rational function coefficients. A singular point x0 of (S) is called an apparent singularity if there is a basis of formal solutions of (S) which are holomorphic in a neighborhood of x0. In this talk we shall present a new algorithm which, given a system of the form (S), detects apparent singularities and constructs an equivalent system (S') with rational coefficients, such that every singularity of (S') is a singularity of (S) that is not apparent. Our method can, in particular, be applied to the companion system of any linear differential equation with arbitrary order n . We thus have an alternative method to the standard methods for removing apparent singularities of linear differential operators. We shall compare our method to the one designed for operators and we shall show some applications and examples of computation.
We define simultaneously row- and column-reduced forms of higher-order linear differential systems with power series coefficients and give two algorithms for their computation. This extends previous work by Barkatou, El Bacha and Pflügel on second-order systems to arbitrary orders. We also show that the algorithm can be used to compute Two-Sided Block-Popov forms. Finally we show how a simultaneously row- and column-reduced form can be used to transform a given higher-order input system into a first-order system.
Joint work with Barkatou, El Bacha and Pflügel
(Session chair: J. Middeke)
The Chow form for an irreducible differential variety is defined and most of the properties of the Chow form in the algebraic case are extended to its differential counterpart. Furthermore, the generalized differential Chow form is defined and its properties are proved. As an application, the sparse differential resultant of n+1 essential differential polynomials in n variables is defined and an algorithm to compute the sparse resultant is presented, which is single exponential in terms of the order, the number of variables, and the size of the differential polynomials.
The development of differential elimination techniques similar to the algebraic existing ones (Groebner basis and multivariate resultants) is an active field of research. Given a system P of n linear ordinary differential polynomial parametric equations (linear DPPEs) in n-1 differential parameters, we proved that, if nonzero a differential resultant gives the implicit equation of P. Unfortunately, differential resultants often vanish under specialization. Motivated by Canny's method and its generalizations, we consider now a linear perturbation of P and use it to give an algorithm to decide if the dimension of the implicit ideal of P is n-1 and, in the affirmative case, obtain the implicit equation of P.
A finite number of rational functions are compatible if they satisfy the compatibility conditions of a first-order linear functional system involving differential, shift and q-shift operators. We present a theorem that describes the structure of compatible rational functions. The theorem enables us to decompose a solution of such a system as a product of a rational function, several symbolic powers, a hyperexponential function, a hypergeometric term, and a q-hypergeometric term. We outline an algorithm for computing this product, and discuss how to determine the algebraic dependence of hyperexponential-hypergeometric elements. This is joint work with Shaoshi Chen, Royong Feng, and Guofeng Fu.
(Session chair: F. Schwarz)
(Session chair: A. Levin)
The purpose of the talk is the constructive study of the concept of purity filtration of a differential module introduced in algebraic analysis and the theory of D-modules. The purity filtration is a natural filtration of a differential module defined by its submodules formed by its elements of codimension (or grade) at least r.
The purity filtration was studied by Björk [Bjork1,Bjork2] using spectral sequences, by Sato and Kashiwara [Kashiwara,Sato] using associated cohomology and, more recently, by Pommaret [Pom1,Pom2] using modified Spencer forms. Moreover, in a recent “tour de force”, Barakat was able to implement the computation of the corresponding spectral sequences [Barakat] in a GAP 4 package called homalg [homalg], which gives one a way to compute the purity filtration of a differential module.
In this talk, we show how the purity filtration can be simply characterized by means of basic concepts and tools of module theory and homological algebra, which avoids the use of sophisticated homological algebra concepts such as spectral sequences, associated cohomology and Spencer cohomology. Moreover, an effective algorithm for the computation of the purity filtration is explained [Q1,Q2] and illustrated by means of its implementation in the Maple package PurityFiltration built upon OreModules [OreModules]. We also use the computation of the purity filtration of a differential module to show that every linear system of partial differential equations is equivalent to a particular block-triangular linear system of partial equations, which allows an integration of the system in cascade by solving equidimensional homogeneous linear systems [Q1,Q2]. We show that the PurityFiltration package can be used to find closed-form solutions of many over-/under-determined linear systems of partial differential equations which cannot be integrated by Maple. Finally, we explain interesting features of our algorithm using its recent implementation in the AbelianSystems package of homalg, developed for abelian categories in collaboration with Barakat, which allows us to start investigating the purity filtration of linear systems over non-regular Auslander rings appearing, for instance, in algebraic geometry.
Fifty years ago D.C. Spencer invented the first order operator now wearing his name in order to bring the formal study of systems of ordinary differential (OD) or partial differential (PD) equations to that of equivalent first order systems. However, despite its importance, the Spencer operator is rarely used in mathematics today and, up to our knowledge, has never been used in engineering applications.
We shall first recall briefly its definition, both in the framework of systems of OD/PD equations and in the framework of differential modules, and then provide a series of explicit and striking applications with explicit experiments.
In a rough way, our main goal is to prove that the use of the Spencer operator constitutes the common secret of the three following famous books published about at the same time in the beginning of the last century, though they do not seem to have anything in common at first sight as they are successively dealing with elasticity theory, commutative algebra, electromagnetism and general relativity:
(Session chair: E. Mansfield)
The need for solving linear inhomogeneous partial differential equations (pde's) arises when homogeneous equations that are reducible but not completely reducible are solved. To this end a new method for solving linear inhomogeneous pde's is described. Lagrange's variation-of-constants method for solving linear inhomogeneous ordinary differential equations (ode's) is replaced by a method based on the Loewy decomposition of the corresponding homogeneous equation. It uses only properties of the equations and not of its solutions. As a consequence it has the advantage that it may be generalized for pde's. It is applied to equations of second order in two independent variables, and to a certain system of third-order pde's. All possible linear inhomogeneous pde's are covered that may occur when third-order linear homogeneous pde's in two independent variables are solved.
We present a geometric method for computing rational general solutions of an algebraic ODE of order 1, whose corresponding algebraic equation defines a rational surface. Since rationality of algebraic surfaces is preserved under a birational mapping, it makes sense to consider birational mappings which are compatible with the set of rational integral curves. In this sense, parametrizable ODEs of order 1 are related to each other via such a birational transformation. We present an affine linear transformation having this property.
A decision algorithm for finding elementary integrals of transcendental Liouvillian functions will be outlined. Parameters that are linearly involved in the integrand can also be solved for, which can be used to find linear relations for definite parameter integrals. Examples of indefinite and definite integrals which can be handled will be given.
(Session chair: G. Labahn)