Organization
Topics
Differential and difference Galois theories of linear differential
and difference equations with parameters
Applications to differential and difference
algebraic dependencies
among solutions of differential and difference equations
Differential and difference elimination
Interactions with computational aspects of differential and difference equations
Interactions with traditional areas of symbolic computation
Related events
July 7, 8, and 14, 2014: lectures at the Kolchin Seminar in Differential Algebra
Financial support
Supported in part by the National Science Foundation grant DMS1413859
Talks
Videos of the talks

Carlos Arreche
Title: Computing differential Galois groups of parameterized secondorder linear differential equations
Abstract:
We describe recent algorithms to compute the differential Galois group G
associated to a parameterized secondorder homogeneous linear
differential equation with respect to d/dx, with coefficients lie in the
field F(x) of rational functions in x with coefficients in a partial
differential field F of characteristic zero. As an application of these
algorithms, we present a set of criteria to decide the differential
transcendence of the solutions with respect to the parametric
derivations on F.
Video

Moulay Barkatou
Title: A direct algorithm for computing ksimple forms of firstorder linear differential systems
Abstract:
Most algorithms for computing solutions of linear ordinary differential
equations proceed by investigating the singularities of the coefficients
to obtain information on the singularities of the solutions. For
systems, this is more difficult than for scalar equations and a main
tool for computing the local exponents and the nonramified local
exponential parts in this case is Moser and superreduction. From
superreduced forms, one can compute all the integer slopes of the
Newtonpolygon and determine the corresponding Newton polynomials.
Moser and superreduced forms are also used in the computation of the
formal solutions of the system around a singularity and in the
computation of the global solutions such as the rational solutions and
the exponential solutions.
However, solving some of these problems requires only weaker forms
called ksimple forms. In this talk, we present a direct (i.e.,
without computing first a superreduced form) algorithm for computing
ksimple forms of firstorder differential systems at x=0 with
coefficients from C((x)), where C is a field of constants. We give the
arithmetic complexity of our algorithm which has been implemented in
Maple and we illustrate it with some examples. Finally, we show how
using this algorithm one can find the formal invariants of the system at
x=0.
Video

James Freitag
Title: Bounding the size of a finite differential algebraic variety
Abstract: Given a collection of differential polynomials, f_1,.., f_n,
suppose that their common solution set is finite. In the ordinary case,
Hrushovski and Pillay showed how to get effective bounds for the size
of this finite solution set in terms of the orders and degrees of the
differential polynomials. This result is at the differential algebraic
heart of various applications of differential algebra to problems in
diophantine geometry (due to Hrushovski, Pillay and recent results of
Freitag and Scanlon). The proof of the upper bound is algorithmic, and
reveals how the problem is related to socalled geometric axioms for
differentially closed fields. The upper bound eventually comes from
intersection theory in arc spaces. We will explain how to extend this
work to the partial case, which is more involved, essentially due to the
coherence condition for differential polynomials. If time permits,
recent diophantine applications related to the the AndreOort conjecture
will be given. This is joint work with Omar Leon Sanchez.
Video

XiaoShan Gao
Title: Binomial difference ideal and toric difference variety
Abstract: The concepts of binomial difference ideals and toric
difference varieties are defined and their properties are proved.
Two canonical representations for Laurent binomial difference ideals
are given using the reduced Groebner basis of Z[x] lattices and
regular and coherent difference ascending chains, respectively.
Criteria for a Laurent binomial difference ideal to be reflexive,
prime, perfect, and toric are given in terms of their support
lattices which are Z[x] lattices. The reflexive and perfect closures
of a Laurent binomial difference ideal are shown to be binomial.
Four equivalent definitions for toric difference varieties are
presented. Finally, algorithms are given to check whether a given
Laurent binomial difference ideal I is reflexive, prime, perfect, or
toric, and in the negative case, to compute the reflexive and
perfect closures of I. An algorithm is given to decompose a finitely
generated perfect binomial difference ideal as the intersection of
reflexive prime binomial difference ideals.
Video

Florian Heiderich
Title: Towards a noncommutative PicardVessiot theory
Abstract: André unified the Galois theories of linear differential
equations and of linear difference equations. This talk aims to explain
a more general approach to a Galois theory of linear functional
equations involving a wide class of operators not only containing
derivations and automorphisms, but also skewderivations.
Video

Maximilian Jaroschek
Title: Radicals of Ore polynomials
Abstract: We give a comprehensible algorithm to compute the radical of an Ore
operator. Given an operator P, we find another operator L and a positive
integer k such that P is the kth power of L and k is maximal among all
integers for which such an operator L exists. Furthermore we discuss
possible extensions of this procedure to identify operators of the form
A*P where P is the kth power of some operator L and to derive a reduced
operator A*L.
Video

Gabriela Jeronimo
Title: Effective differential Lüroth theorem
Abstract: Let F be a differential field of characteristic 0 and L the
field of differential rational functions in a single indeterminate
u. The differential Lüroth theorem proved by Ritt and extended by
Kolchin states that for any differential subfield G of L there exists
v in G such that G is the field of differential rational functions in v.
The talk will focus on effectivity aspects of this result. More
precisely: given nonconstant rational functions v_1 ,..., v_n in
L such that G is the field of differential rantional functions in
v_1,...,v_n, we will give upper bounds for the total order and degree of
a Luroth generator v of the extension G/F in terms of the number and
the maximum order and degree of the given generators of G.
Our approach combines elements of Ritt's and Kolchin's proofs with
estimations concerning the order and the differentiation index of
differential ideals. As a byproduct, we will show that our techniques
enable the computation of a Lüroth generator by dealing with a
polynomial ideal in a polynomial ring in finitely many variables.
This is joint work with Lisi D'Alfonso and Pablo Solerno.
Video

Lourdes Juan
On the integration of algebraic functions: computing the logarithmic part
Bronstein developed a complete algorithm to compute the logarithmic part
of an integral of a function that lies in a tower of transcendental
elementary extensions. However, computing the logarithmic part in an
algebraic extension has remained difficult and challenging. In his PhD
dissertation, Brian Miller, building on the work of Manuel Kauers,
developed a method to compute the logarithmic part when the function
lies in a tower of transcendental elementary extensions followed by an
algebraic extension. The method uses Grobner bases and primary
decomposition. In this talk we will discuss Millerâ€™s work and work in
progress to produce a complete algorithm for some particular cases of
algebraic functions.
Video

Irina Kogan
Title: Differential algebra of invariants and invariant variational calculus
Abstract: Systems of differential equations and variational problems
arising in geometry and physics often admit a group of symmetries. As
was first recognized by S. Lie, these problems can be rewritten in terms
of groupinvariant objects: differential invariants, invariant
differential forms, and invariant differential operators. Differential
invariants and invariant differential operators constitute a
differential algebra with often nontrivial but computable structure.
It is desirable from both computational and theoretical points of view
to study invariant problems in terms of invariant differential algebra.
In this talk, we will describe how differential algebra of invariants
can be constructed and show applications to variational
calculus.
Video

George Labahn
Title: Convolution integrals of holonomic functions
Abstract: In this talk we describe a procedure for determining closed
form solutions of convolution integrals of holonomic functions. Because
such integrals require knowing strips of convergence in the complex
plane determining closed form solutions requires a mix of both algebra
and analysis. We describe the tools used from complex analysis,
particularly Mellin transforms, analytic continuations and distributions
which all play vital roles in the analysis.
This is joint work with Jason Peasgood (Waterloo) and Bruno Salvy
(France).
Updated
video

Markus LangeHegermann
Title: Counting solutions of differential equations
Abstract: The aim of this talk is a quantitative analysis of the solution set of a
system of differential equations. We generalize Kolchin's differential
dimension polynomial and its properties from prime differential ideals
to characterizable differential ideals. This makes the dimension
polynomial more accessible for algorithms. In certain applications, an
even more detailed quantitative description of differential equations is
needed. Therefore, we introduce the differential counting polynomial, a
generalization of the differential dimension polynomial. The tools used
in this talk are the decomposition algorithms by J.M. Thomas.
Video

Francois Lemaire
Title : New development and application of integration of differential fractions
Abstract :
The publication "On the Integration of Differential Fractions" published
at ISSAC'13 (Boulier, Lemaire, Regensburger, Rosenkranz) introduces the
differential fractions (which are
quotients of differential polynomials) and presents algorithms
for representing such fractions. In particular, a differential
fraction F can be decomposed into F = G + dH/dx (where G and H are
differential fractions and d/dx designates
the total derivative w.r.t. x). I will present recent developments
concerning the canonicity of such a decomposition, as well as
numerical applications of the decomposition in the context of
parameters estimation.
Video

Omar Leon Sanchez
Title: Parametrized logarithmic equations and their Galois theory
Abstract: I will talk about (parametrized) differential Dgroups which are
not necessarily defined over a field of constants. Then, I will present
the foundational results on the Galois theory of logarithmic differential
equations in such groups. I will also discuss two natural nonlinear
examples of such equations which are not defined over the
constants.
Video

Alexander Levin
Title: Generalized Groebner bases and dimension polynomials of modules over some finitely generated noncommutative algebras
Abstract: I will present a generalized Groebner basis method in free
modules
over finitely generated noncommutative algebras of a certain class
that includes, in particular, algebras of differencedifferential
operators, Ore algebras, quantized (and classical) Weyl algebras. I
will show the existence and give methods of computation of univariate
and multivariate dimension polynomials associated with systems of
generators of finitely generated modules over such algebras. I will also
determine invariants of the dimension polynomials, that is, numerical
characteristics carried by these polynomials that do not depend on the
choice of the corresponding systems of module generators.
Video

Suzy S. Maddah
Title:
Formal Solutions of Completely Integrable Pfaffian Systems with Normal Crossings
Abstract:
Abstract: In this talk, we are interested in the formal reduction of
the socalled completely integrable Pfaffian systems with normal
crossings, i.e. the class of linear systems of partial differential
equations. Pfaffian systems arise in many applications including
the studies of aerospace and celestial mechanics (see, e.g., Awane et
al.' 2000 and Broucke' 1978). By far the most important for applications
are those with normal crossings (Novikov et al.' 2004). The
theoretical results by Charrière et al.'19801981 and van den Essen et
al.' 1982, give the existence and properties of a fundamental
matrix of formal solutons. However, the formal reduction, i.e. the
algorithmic procedure that computes the transformation which takes the
system into its canonical form so that formal solutions can be
constructed, is a question of another nature. Clearly, the
particular univariate case corresponds to singular linear systems of
ordinary differential equations which have been studied extensively
(see, e.g., Balser' 2000, Wasow' 1965, and references therein). Moreover,
unlike the multivariate case, algorithms to related problems leading to
the construction of the formal solutions have been developed by various
authors (see, e.g., Barkatou et al.' 19972009, and references therein).
We present an algorithm to construct a fundamental matrix of
formal solutions of completely integrable Pfaffian systems with normal
crossings in two variables. Our algorithm is based on associating to
the Pfaffian system a singular linear system of ordinary differential
equations from which its formal invariants can be efficiently derived.
We then discuss the extension of this algorithm to the multivariate case.
Our algorithm builds upon the package ISOLDE and is implemented in the
computer algebra system Maple.
Video

Annette Maier
Title: Parameterized differential equations and patching
Abstract: We consider linear parameterized differential equations over
k((t))(x) with parameter t such as d_x(y)=ty/x. The parameterized Galois
group of such an equation is a linear differential algebraic group over
k((t)), i.e., a subgroup of GL_n given by differential d_talgebraic
equations. In the example above, the parameterized Galois group equals
G={c nonzero  d_t^2(c)cd_t(c)^2=0}. The inverse problem asks which
linear differential algebraic groups occur as parameterized Galois
groups. In the talk, I will explain how algebraic patching methods can
be applied to realize a certain class of groups as parameterized Galois
groups over k((t))(x).
Video

Alice Medvedev
Title: Dimensions of differencealgebraic groups
Abstract: Modeltheoretic dimensions (Lascar rank, Morley rank) of
groups defined by difference equations of the form x in G and
sigma^m(x) = f(x) for some algebraic group G and some
algebraic group morphism f : G> sigma^m(G) are
relatively easy to compute. I will discuss two interesting and useful
notions of the limit of these dimensions as m tends to
infinity.
Video

Sergey Paramonov
Title: Undecidability of the uniqueness testing problem for analytic solutions of PLDE with boundary conditions
Abstract: We consider linear partial differential equations with
polynomial coefficients and prove
algorithmic undecidability of the following problem: to test whether a
given equation of considered form has no more than one solution that is
analytic on an open region and that satisfies some fixed boundary
conditions. It is assumed that a polynomial which vanishes at each of
points of the region bound is known.

Julien Roques
Title: Galois groups of difference equations on elliptic curves
Abstract: The main purpose of this talk is to compute the Galois groups
of some difference equations of order two on elliptic curves, such as
discrete Lamé equations.

Michael Wibmer
Title: A JordanHölder theorem for difference algebraic groups
Abstract: Difference algebraic groups, i.e., groups defined by algebraic
difference equations occur as the Galois groups of linear differential
and difference equations depending on a discrete parameter. In this talk
I will introduce difference algebraic groups and explain some of their
basic properties. In particular, I will present an analog of the
JordanHölder theorem for difference algebraic groups.
Video

Meng Zhou
Title: On the termination of algorithm for computing relative Gröbner bases and differencedifferential dimension polynomials
Abstract: Relative Gröbner bases were introduced by Zhou and
Winkler(2008) in order to compute bivariate dimension polynomials in
differencedifferential modules. The algorithm for computing relative
Gröbner bases and bivariate dimension polynomials also were
presented in Zhou and Winkler(2008). Christian Dönch (2010) gave
a Maple software of the algorithm. By now it is used as the main tool
for the algorithmic computation of bivariate dimension polynomials in
differencedifferential modules.
Recently Christian Dönch(2013) presented an example pointing out
the algorithm does not terminate in some case. From the counterexample
Dönch pointed out that it is questionable whether a relative
Gröbner basis always exists. Also it is uncertain whether the
algorithm for computing bivariate dimension polynomials in
differencedifferential modules terminates.
In this paper we improve the results of Zhou and Winkler(2008) about
relative Gröbner bases. We introduce the concept of
differencedifferential degree compatibility on generalized term orders.
Then we prove that in the process of the algorithm the polynomials with
higher and higher degree wouldn't be produced, if the term orders
"<" and "<'" are differencedifferential degree compatibility. So
we present a condition on the generalized orders and prove that under
the condition the algorithm for computing relative Gröbner bases will
terminate. Also the relative Gröbner bases exist under the condition.
Finally we prove the algorithm for computation of the bivariate
dimension polynomials in differencedifferential modules terminates.