Algebraic and Algorithmic Aspects of Differential and Integral Operators Session

ACA'08

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

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

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:

- Factorization of Differential/Integral Operators
- Constructive Methods for D-Modules
- Linear Initial/Boundary Value Problems and Green's Operators
- Rota-Baxter Algebras
- Symbolic Computation with Differential Polynomials
- Initial/Boundary Value Problems for Nonlinear Differential Equations

*
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.
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**Slides.** For download here.

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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.
*

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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.
*

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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.
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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.
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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).
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**Slides.** For download here.

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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.
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**Slides.** For download here.

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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.
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**Slides.** For download here.

*
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.
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**Slides.** For download here.

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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.
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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.
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**Slides.** For download here.

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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).
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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.
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**Slides.** For download here.

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Abstract. See the pdf.
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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.
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**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 |