Algebraic and Algorithmic Differential and Integral Operator Session

ACA 2011

Houston, Texas, USA, 27-30 June 2011

## Organizers

Markus Rosenkranz (University of Kent, SMSAS, Canterbury, United Kingdom)
Georg Regensburger (INRIA Saclay, Île de France, L2S, Gif-sur-Yvette, France)

## Scope

The algebraic/symbolic treatment of differential equations is a flourishing field, branching out in a variety of subfields committed to different approaches. In this session, we want to give special emphasis to the operator perspective of both the underlying differential operators and various associated integral operators (e.g. as Green's operators for initial/boundary value problems).

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

## Scheduled Talks

#### Islam Boussaada (Laboratoire des Signaux et Systèmes, Supélec, France) 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.

#### Alexei Cheviakov (University of Saskatchewan, Canada) 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.

#### Christoph Koutschan (Johannes Kepler University, Austria) 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.

#### Alexander Levin (Catholic University of America, United States) 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.

#### Anton Leykin (Georgia Institute of Technology, United States) 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 .

#### Loredana Tec (Kepler University Linz, Austria) 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.)

#### Hiroshi Umemura (Nagoya University, Japan) 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 .

#### Franz Winkler (Kepler University Linz, Austria) 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.