Algebraic and Algorithmic Aspects of Differential and Integral Operators Session

ACA 2012

Sofia, Bulgaria, 25-28 June, 2012

Thomas Cluzeau (University of Limoges, CNRS, XLIM UMR 7252, DMI, Limoges, France)

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

Markus Rosenkranz (University of Kent, SMSAS, Canterbury, United Kingdom)

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 AADIOS '11 , AADIOS '10 , AADIOS '09, and AADIOS '08.

See here for an MCS double Special Issue on previous AADIOS sessions.

**Abstract. ** The product of polynomials and the product of matrices are two of the most basic operations in mathematics;
the study of their computational complexity is central in computer science.
In this talk, we will be interested in the computational complexity of multiplying two linear differential operators.
These algebraic objects encode linear differential equations, and form a non-commutative ring that shares many properties
with the commutative ring of usual polynomials. Yet, the algorithmic study of linear differential operators is currently
much less advanced than in the polynomial case: the complexity of multiplication has been addressed only recently,
but not completely solved. The aim of the present work is to make a step towards filling this gap,
and to solve an open question raised by van der Hoeven. This work is joint with Alin Bostan and Joris van der Hoeven.

**Abstract. ** We study tight bounds and fast algorithms for LCLMs of several
linear differential operators with polynomial coefficients. We analyse the
arithmetic complexity of existing algorithms for LCLMs, as well as the size of
their outputs. We propose a new algorithm that recasts the LCLM computation in
a linear algebra problem on a polynomial matrix. This algorithm yields sharp
bounds on the coefficient degrees of the LCLM, improving by one order of
magnitude the best bounds obtained using previous algorithms. The complexity
of the new algorithm is almost optimal, in the sense that it nearly matches
the arithmetic size of the output. Joint work with Frédéric Chyzak, Ziming Li
and Bruno Salvy.

**Slides.** For download here.

**Abstract. **For download here.

**Abstract. **For download here.

**Slides.** For download here.

**Abstract. **For download here.

**Abstract. **For download here.

**Abstract. **For download here.

**Abstract. **For download here.

**Abstract. **A finite number of rational functions are compatible if they satisfy the
compatibility conditions of a first-order linear functional system involving differential and shift operators.
In this talk, 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 and a hypergeometric term.
Some applications are also presented in this talk. This is joint work with Shaoshi Chen, Guofeng Fu and Ziming Li.

**Abstract. **In this talk, we first introduce the concept of Laurent
differentially essential systems and give a criterion for Laurent
differentially essential systems in terms of their supports. Then
the sparse differential resultant for a Laurent differentially
essential system is defined and its basic properties are proved. In
particular, order and degree bounds for the sparse differential
resultant are given. Based on these bounds, an algorithm to compute
the sparse differential resultant is proposed, which is single
exponential in terms of the order, the number of variables, and the
size of the Laurent differential system. Detailes of the results can be found in
W. Li, C.M. Yuan, X.S. Gao, Sparse Differential Resultant for Laurent Differential Polynomials.
arxiv-1111.1084v2, 62 pages, 2011.

**Abstract. **For download here.

**Slides.** For download here.

**Abstract. **
The concept of integro-differential algebra has been introduced
recently in the study of boundary problems for differential equations. Its
free objects can be obtained from those of the closely related differential
Rota-Baxter algebra by taking quotients. We study their explicit
construction in terms of integro-differential polynomials. Joint work with
Georg Regensburger and Markus Rosenkranz.

**Abstract. **For download here.

**Abstract. **
We present two new closure properties for q-holonomic sequences,
namely twisting by complex roots of unity and raising q to a rational
power. The proofs are constructive, work in the multivariate setting
of d-finite sequences and are implemented in our Mathematica package
HolonomicFunctions. The results are illustrated by twisting natural
q-holonomic sequences which appear in quantum topology, namely the
colored Jones polynomial of pretzel knots and twist knots. The
recurrence of the twisted colored Jones polynomial can be used to
compute the asymptotics of the Kashaev invariant of a knot at an
arbitrary complex root of unity. This is joint work with Stavros
Garoufalidis.

**Slides.** For download here.

**Abstract. **For download here.

**Abstract. **For download here.

**Abstract. **
The correspondence between solution sets of systems of algebraic
equations and radical ideals of the affine coordinate ring is
fundamental for algebraic geometry. This talk discusses aspects of
an analogous correspondence between systems of polynomial differential
equations and their analytic solutions. Implicitization problems for
certain families of analytic functions are approached in different
generality. While the linear case is understood to a large extent,
the non-linear case requires new algorithmic methods, e.g., the use of
differential inequations, as proposed by J. M. Thomas in the 1930s.

**Abstract. **
The basic idea that integration is "somehow" a division by the differential
operator has haunted mathematicians ever since the time of Oliver Heaviside.
Put on a firm algebraic basis by Jan Mikusinski, the operational calculus has now been vastly extended and generalized,
notably by Ivan Dimovski and his schol. In this talk we sketch a new approach to algebraic operational calculi.
Treating Green's operators on a par with the standard integral operator (Duhamel convolution),
we build up a localization of a suitable operator ring that can be made to act on an algebraic space of hyperfunctions.
Joint work with A. Korporal. Earlier work on this topic was pursued in collaboration with G. Regensburger.

**Slides.** For download here.

**Abstract. **For download here.

**Abstract. **
We present a new approach for solving boundary value problems for linear systems of differential equations
allowing two-/multi-point as well as arbitrary Stieltjes conditions as in the scalar case.
For expressing differential operators, boundary conditions,
and Green's operators, we employ the algebra of integro-differential operators.

**Slides.** For download here.

**Abstract. **For download here.

**Abstract. **For download here.

**Abstract. **
In this talk, an intersection theory for generic differential polynomials
is presented. The intersection of an irreducible differential variety of dimension
d and order h with a generic differential hypersurface of order s is
shown to be an irreducible variety of dimension d-1 and order h+s. As a
consequence, the dimension conjecture for generic differential polynomials is
proved. Based on the intersection theory, 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 of the generalized differential Chow form, the
differential resultant of n + 1 generic differential polynomials in n variables
is defined and properties similar to that of the Sylvester resultant of two
univariate polynomials are proved.

**Abstract. **For download here.