Algebraic and Algorithmic Differential and Integral Operator Session

ACA 2011

Houston, Texas, USA, 27-30 June 2011

Georg Regensburger (INRIA Saclay, Île de France, L2S, Gif-sur-Yvette, France)

In particular, we invite contributions in line with the following topics:

- Symbolic Computation for Operator Algebras
- Factorization of Differential/Integral Operators
- Linear Boundary Problems and Green's Operators
- Initial Value Problems for Differential Equations
- Symbolic Integration and Differential Galois Theory
- Symbolic Operator Calculi
- Algorithmic D-Module Theory
- Rota-Baxter Algebra
- Differential Algebra
- Discrete Analogs of the above
- Software Aspects of the above

Please see also our previous ACA sessions AADIOS08 and AADIOS09 and AADIOS10. We have prepared an MCS Special Issue based on these sessions.

Linearization of Planar Systems of ODE

*
Abstract. We consider a three parameter homogeneous
perturbations (of arbitrary degree n>2) of the linear canonical
isochronous center x'=-y, y'=x. Using the CAR algorithm, we obtain a
characterization of linearizable systems. Moreover, a careful
inspection of the necessary conditions allows us to compute explicitly
the depth of the linearizability problem. The obtained depth
simplifies the linearisability investigation for additional
homogeneous perturbations of arbitrary degree m>2n-1.
*

Unfortunately, this talk was cancelled due to visa problems.

Computation of symmetries and exact invariant solutions of differential equations: Symbolic software and examples

*
Abstract. I will briefly discuss the idea and computation
algorithm for Lie symmetries and symmetry-invariant solutions of
ordinary and partial differential equations, and then demonstrate
Maple-based symbolic software that implements these algorithms. Run
examples will be presented where symmetries and symmetric reductions
are obtained for a number of physically important systems of
differential equations.
*

Unfortunately, this talk was cancelled due to illness of the speaker.

Lattice Green's Functions

*
Abstract. We study the lattice Green's functions of the
face-centered cubic lattice (fcc) in up to six dimensions. We give
computer algebra proofs of results that were conjectured by Guttmann
and Broadhurst for the four- and five-dimensional fcc
lattices. Additionally we derive a differential equation for the
lattice Green's function of the six-dimensional fcc lattice, a result
that was not believed to be achievable with current computer hardware.
*

Please see the slides here.

Methods of Computation and Invariants of Difference-Differential Dimension Polynomials

*
Abstract. Difference-differential dimension polynomials are
Hilbert-type polynomials associated with finite systems of generators
of modules over rings of difference-differential operators and
Kolchin-type polynomials associated with difference-differential field
extensions and systems of algebraic partial difference-differential
equations. We discuss methods of computations of univariate and
multivariate difference-differential polynomials of both types and
consider invariants of such polynomials, that is, characteristics of a
difference-differential module or a difference-differential field
extension, which are carried by its dimension polynomial and which do
not depend on the choice of the system of generators this polynomial
is associated with. We also present a new interpretation of some of
these invariants.
*

Please see the slides here.

Computing localizations iteratively

*
Abstract. The localization $R_f$ of a polynomial ring $R$, where $f\in
R$ can be computed as a holonomic $D$-module using Gröbner bases in
the algebra of linear differential operators with polynomial
coefficients $D$. We develop an iterative algorithm that limits the
use of Gröbner to $R$ and produces a sequence of $D$-modules
stabilizing at $R_f$. (Joint work with Francisco-Jesús Castro-Jiménez)
*

Please see the slides here .

Symbolic Boundary Problems in the Theorema Framework of General Polynomial Domains

*
Abstract. In this talk, we present a symbolic framework for working with general
polynomial domains, in the sense of monoid algebras and universal algebra.
This framework is developed in Theorema, an integrated
environment for doing mathematics. As a particular application, we give
an overview of the integro-differential operators used for a symbolic
treatment of linear boundary problems. We consider the case of the
integro-differential operators for ordinary differential equations, as
well as some experimental steps for partial differential equations. We
conclude with some examples to illustrate this approach, using our
generic implementation in the Theorema system. (Joint work with
Markus Rosenkranz, Georg Regensburger and Bruno Buchberger.)
*

Please see the slides here.

Soliton theory is Abelian

*
Abstract.We apply general Galois theory of differential
equations to soliton theory. The observation from Galois theory tells
us that Galois group of KP-hierarchy is Abelian. This shows that
Galois theoretically the soliton equations look integrable.
*

Please see the slides here .

Algebra + Geometry ==> Differential Equation Solving

*
Abstract. Consider an autonomous algebraic ODE of the form
$F(y,y')=0$, where $F$ is a bivariate polynomial. We can think of $F$
as defining a plane algebraic curve. If this curve admits a rational
parametrization, then we can determine whether the ODE has a rational
general solution. Based on degree bounds for such parametrizations by
Sendra and Winkler, Feng and Gao have described an algorithm for this
problem.*

*
Here we extend this investigation to the case of a non-autonomous
algebraic ODE of the form $F(x,y,y')=0$. The tri-variate polynomial
$F(x,y,z)$ defines an algebraic surface, which we assume to admit a
rational parametrization. Based on such a parametrization and on
knowledge about a degree bound for general rational solutions, we can
determine the existence of a general rational solution, and, in the
positive case, also compute one. This method depends crucially on the
determination of rational invariant algebraic curves. We also relate
rational general solutions of algebraic ODEs to rational first
integrals.
*

This talk was held as an invited ACA Plenary Talk! Please see the slides here.