Algebraic and Algorithmic Aspects of Differential and Integral Operators Session


RISC-Linz, Castle of Hagenberg, Austria, July 27-30, 2008


Markus Rosenkranz (Austrian Academy of Sciences, RICAM, Linz, Austria)
Georg Regensburger (Austrian Academy of Sciences, RICAM, Linz, Austria)


The algebraic treatment of differential equations is a well-established field with close ties to the symbolic community (see also the ACA Session "Symbolic Symmetry Analysis and its Applications"). Algebraic methods often commence from an operator perspective on the underlying differential equations, e.g. in D-Module theory or in factoring linear differential operators (ODE/PDE, scalar/vector). On the other hand, integral operators have as yet received comparably little attention in an algebraic setting. In the context of linear differential equations, they arise naturally as Green's operators for initial/boundary value problems.

In this session, we would like to examine various relations between differential and integral operators. To this end, we want to bring together the following topics and communities:

Recent connections between these topics include algebraic structures combining derivations with integrals (e.g. differential Rota-Baxter algebras) and the correspondence between factorizations of differential and integral operators (e.g. by splitting their boundary value problems). We hope that we will have a stimulating discussion that may lead to further interrelations.


Wolf Daniel Andres: Various algorithms for the computation of Bernstein-Sato polynomial (Short Talk)

Abstract. We describe Noro's algorithm for computing the b-function, also called the Bernstein-Sato polynomial, and its implementation in the computer algebra system SINGULAR. In doing so, we are especially interested in the b-functions of hyperplane arrangements. The knowledge of b-function is important on its own and is also vital for many applications in D-module theory. We will systematically compare different methods (preliminary Noro's) and their implementations. Additionally, we will discuss other approaches and problems as well.

Slides. For download here.

Moulay Barkatou: Hypergeometric Solutions of Linear Difference Systems [CANCELED]

Abstract. We will present a direct algorithm for computing hypergeometric solutions of systems of linear difference equations and discuss applications to finding second order right-hand actors of a given linear difference operator with polynomial coefficients. This talk is based on a joint work with Mark van Hoeij.

Thomas Cluzeau: On algebraic simplifications of linear functional systems

Abstract. Within a constructive homological algebra approach, we study the factorization and decomposition problems for a class of linear functional (determined, over-determined, under-determined) systems. Using the concept of Ore algebras of functional operators (e.g., ordinary/partial differential operators, shift operators, time-delay operators), we first concentrate on the computation of morphisms from a finitely presented left module M over an Ore algebra to another one M', where M (resp., M') is a module intrinsically associated with the linear functional system R y=0 (resp., R' z=0). These morphisms define applications sending solutions of the system R' z=0 to solutions of R y=0.

We explicitly characterize the kernel, image, cokernel and coimage of a general morphism. We then show that the existence of a non-injective endomorphism of the module M is equivalent to the existence of a non-trivial factorization R=R2 R1 of the system matrix R. The corresponding system can then be integrated ``in cascade". Under certain conditions, we also show that the system R y=0 is equivalent to a system R' z=0, where R' is a block-triangular matrix of the same size as R.

We show that the existence of idempotents of the endomorphism ring of the module M allows us to reduce the integration of the system R y=0 to the integration of two independent systems R1 y1=0 and R2 y2=0. Furthermore, we prove that, under certain conditions, idempotents provide decompositions of the system R y=0, i.e., they allow us to compute an equivalent system R' z=0, where R' is a block-diagonal matrix of the same size as R. Applications of these results in mathematical physics and control theory are given.

Finally, the different algorithms of the paper are implemented in a Maple package called OreMorphisms based on the library OreModules and freely available here. This is a joint work with Alban Quadrat (INRIA Sophia-Antipolis APICS Project).

Slides. For download here.

Li Guo: On differential Rota-Baxter algebras

Abstract. We study an associative algebra that carries a differential type operators and an integral type operator, related by an abstraction of the first fundamental theorem of calculus. Such an algebra is called a Rota-Baxter differential algebra (Guo and Keigher) or an integro-differential algebra (Rosenkranz and Regensburger). We will construct the free objects and study the enumerative properties of their bases.

Slides. For download here.

Christoph Koutschan: Holonomic function identities

Abstract. We consider Gröbner bases in Ore algebras (e.g., mixed Weyl and shift algebras), and show their application to finding and proving identities for holonomic functions. This class includes a wide variety of special functions as well as their integrals and sums.

Slides. For download here.

Viktor Levandovskyy: New algorithms and implementations for Computational D-Module theory

Abstract. In a joint work with Jorge Morales, we describe two recent algorithms, LOT and checkRoot together with the new proof of Briancon-Maisonobe algorithm. We demonstrate our implementation of fundamental algorithms in computational D-module theory in computer algebra system SINGULAR and show the comparison with different implementations, which clearly distinguishes our one. We present first solutions to several computational challenges. Moreover, some applications and future developments will be discussed.

Slides. For download here.

Anton Leykin: Degree bounds for Gröbner bases in algebras of solvable type [CANCELED]

Abstract. In joint work with Matthias Aschenbrenner we have modified the combinatorial approach of Dubé for commutative polynomial rings in order to obtain bounds on the degrees of the elements of Gröbner bases in algebras of the solvable type. These bounds turn out to be doubly-exponential in the number of generators of the algebra as in the commutative case.

Franz Pauer: Gröbner bases with coefficients in rings

Abstract. We first recall different approaches to the theory of Gröbner bases (in commutative polynomial rings and in rings of differential operators) with coefficients in commutative noetherian rings and give some motivations for this theory. Finally a definition of reduced Gröbner basis is proposed.

Slides. For download here.

Georg Regensburger and Markur Rosenkranz: Integro-Differential Algebras as a Natural Setting for Boundary Problems

Abstract. Integro-differential algebras form a natural generalization of differential algebras for stating and solving boundary problems for LODE. The solutions are expressed in terms of integral operators ("Green's operators"), conceived as elements of an algebra of integro-differential operators (generalizing the algebra of differential operators). We define a compositional structure on boundary problems that corresponds to the composition of Green's operators in reverse order. A crucial property of this structure is that every factorization of the differential operator in a boundary problem can be lifted to a factorization of the whole problem (and hence its Green's operator).

In analogy to differential polynomials (and as a basis for constructing extensions), we introduce also the integro-differential polynomials by adjoining an indeterminate to a given integro-differential algebra. For making computations possible, we have set up a canonical simplifier for integro-differential polynomials.

Slides. For download here.

Margarita Spiridonova: Extensions of the Heaviside Algorithm and the Duhamel Principle for Nonlocal Cauchy Problems

Abstract. See the pdf.

Franz Winkler: Relative Gröbner bases and dimension polynomials in difference-differential modules

Abstract. We introduce the notion of relative Gröbner bases w.r.t. to two generalized term orders. Based on this concept of relative Gröbner bases we present a new algorithmic approach to computing the dimension polynomial or Hilbert polynomial for finitely generated difference-differential modules equipped with a natural double filtration.

Slides. For download here.


Sunday, 27 July 2008 Monday, 28 July 2008
13:30-14:00 Cluzeau 17:00-17:30 Winkler
14:00-14:30 Spiridonova 17:30-18:00 Guo
15:00-15:30 Pauer 18:00-18:30 Regensburger
15:30-16:00 Levandovskyy
16:00-16:30 Andres
17:00-17:30 Koutschan