Abstracts
Speaker: Ruyong Feng.
Title: Parallel Differential Telescoping
Abstract:
Creative telescoping is a powerful technique for treating symbolic sums and
integrals in an algorithmic way. This technique now has been applied to many
problems from various areas of mathematics and physics. As a kind of this
technique, differential creative telescoping produces from an integrand a
linear differential operator satisfied by the integral. Motivated by the
direct problem in parametrized differential Galois theory, we introduce the
notion of parallel differential telescoping, which is a natural generalization
of differential creative telescoping for single integrals to integrals of
differential forms. Precisely, let $K=k(t,x_1,\cdots,x_m)$ be the field of
rational functions over $k$ and
$\omega= \sum_{1\leq i_1 < i_2\cdots < i_p\leq n}f_{i_1i_2\cdots i_p}dx_{x_1}dx_{i_2}\cdots dx_{i_p}$
be a differential $p$-form, where the $f_{i_1i_2\cdots i_p}$ are elements in
some differential extension field of $K$. Parallel differential telescoping
produces a nonzero operator $L$ in $k(t)[\partial/\partial_t]$, called a
parallel telescoper, such that
$L(\omega)=\sum_{1\leq i_1 < i_2\cdots < i_p\leq m}L(f_{i_1i_2\cdots i_p})dx_{i_1}dx_{i_2}\cdots dx_{i_p}=d(\eta)$
where $\eta$ is a differential $p-1$-form and $d$ is the exterior derivative
operator with respect to $x_1,\cdots,x_m$.
In this talk, we first recall differential creative telescoping and then give
a brief introduction to parametrized differential Galois theory. After these,
we present a necessary and sufficient condition guaranteeing the existence of
parallel telescopers for differential forms $\omega$ with $f_{i_1i_2\cdots
i_p}$ in the set of differentially finite functions over $K$. When restricting
$f_{i_1i_2\cdots i_p}$ to hyperexponential functions or algebraic functions
over $K$, we present algorithms for deciding whether the operator $L$ exists
or not and algorithms for computing $L$ if it exists. Finally, we discuss some
applications.
This is joint work with Shaoshi Chen, Ziming Li, Michael F. Singer and Stephen Watt.
Speaker: Christoph Koutschan.
Title: Reduction-Based Creative Telescoping for Algebraic Functions
Abstract:
Creative telescoping is a powerful technique to tackle summation and
integration problems symbolically, but it can be computationally very
costly. Many existing algorithms compute two objects, called telescoper and
certificate, but in many applications only the first one is of interest, while
typically the second one is larger in size. In the past few years a new
direction of research was initiated, namely to develop creative telescoping
algorithms that are based on Hermite-type reductions, which avoid the
computation of the certificate and therefore can be more efficient in
practice. We develop a new algorithm to construct minimal telescopers for
algebraic functions, based on Trager's reduction and on a so-called polynomial
reduction. The latter was originally designed for hyperexponential functions
and is now extended to the algebraic case. We view these results as a step
towards a reduction-based creative telescoping algorithm for general holonomic
functions. In this talk, we will also discuss some future directions to
achieve this goal. This is joint work with Shaoshi Chen and Manuel Kauers.
Speaker: Maximilian Jaroschek.
Title: Desingularization of First Order Linear Difference Systems with
Rational Function Coefficients
Abstract:
It is well known that for a first order system of linear difference equations
with rational function coefficients, a solution that is holomorphic in some
left half plane can be analytically continued to a meromorphic solution in the
whole complex plane. The poles stem from the singularities of the rational
function coefficients of the system. Just as for systems of differential
equations, not all of these singularities necessarily lead to poles in a
solution, as they might be what is called removable. In our work, we show how
to detect and remove these singularities and further study the connection
between poles of solutions, removable singularities and the extension of
numerical sequences at these points.
Speaker: Zafeirakis Zafeirakopoulos.
Tentative title: Polyhedra, Symbolic Computation and Applications
Abstract:
The use of polyhedral geometry in the solution of combinatorial, number
theoretic and algebraic problems is gaining momentum in the last decades.
In the first part of this talk, Polyhedral Omega will be presented, an algorithm
combining ideas from polyhedral geometry and the theory of integer partitions.
In the second part we will explore applications and current research on the topic.