MACIS 2015 Session (SS8): Computational theory of differential and difference equations
(RWTH Aachen University, Germany)
(CUNY Queens College and Graduate Center, USA)
(University of Pennsylvania, USA)
Aim and Scope
Symbolic methods for systems of polynomial differential and difference
equations aim at answering such fundamental questions as whether such
systems have solutions and, if yes, then measure the size of the
solution set. It is critical to improve the efficiency of the existing
methods and estimate their computational complexity. Symbolic-numeric
methods aim at finding the answer significantly faster. It is of
special importance to investigate the applicability of such methods to
particular classes of problems.
Topics (including, but not limited to)
- symbolic computation for systems of polynomial differential and difference equations
- symbolic-numeric methods for differential and difference equations
- linear and non-linear differential and difference equations and their symmetries
- differential, difference and integral operators
- applications of differential and difference algebra
- implementation of these algorithms
- complexity estimates of these algorithms
- A short abstract will appear on the conference web page
as soon as accepted,
and the post-conference proceedings will be
published by LNCS.
- A special issue of the journal
Mathematics in Computer Science,
Birkhauser/Springer, will be organized after the
conference by session organizers. REGULAR (not SHORT) papers would be considered for these special issues.
- If you would like to give a talk at MACIS, you need to submit
at least a SHORT paper -- see
for the details.
- The deadline for all submissions is September 1, 2015 -- see
the Call for Papers for the details
- After the meeting,
the submission guideline for a journal special issue
will be communicated to you by the session organizers.
Dimension Polynomials of Intermediate Fields of Inversive Difference Field Extensions
Alexander Levin (The Catholic University of America, USA)
Let K be an inversive difference field, L a finitely generated
inversive difference field extension of K, and F an intermediate
inversive difference field of this extension. We prove the existence
and establish properties and invariants of a numerical polynomial
that describes the filtration of F induced by the natural filtration
of the extension L/K associated with its generators. Then we
introduce concepts of type and dimension of the extension L/K
considering chains of its intermediate fields. Using properties of
dimension polynomials of intermediate fields we obtain relationships
between the type and dimension of L/K and difference birational
invariants of this extension carried by its dimension polynomials.
Finally, we present a generalization of the obtained results to the
case of multivariate dimension polynomials associated with a given
inversive difference field extension and a partition of the basic
set of translations.
A new bound for the existence of differential field extensions
Richard Gustavson (CUNY Graduate Center, USA) and Omar Leon Sanchez (McMaster University, Canada)
We prove a new upper bound for the existence of a differential field extension of a differential field K that is compatible with a given field extension of K. In 2014, Pierce provided an upper bound in terms of lengths of certain antichain sequences of N^m equipped with the product order. This result has had several applications to effective methods in differential algebra such as the effective differential Nullstellensatz problem. Using a new approach involving Macaulay's theorem on the Hilbert function, we produce an improved upper bound.
A "polynomial shifting" trick in differential algebra
Gleb Pogudin (Moscow State University, Russia)
Standard proofs of the primitive element theorem and the Noether
normalization lemma are based on a consideration of "generic
combinations" of initial generators. We propose a differential
counterpart of this argument which we call a "polynomial shifting"
trick. It is an important part of recent proofs of a strengthened
version of Kolchin's primitive element theorem and a differential
analog of the Noether normalization lemma. This trick turned out
to be quite flexible and constructive. We hope that this method
will be useful in dealing with problems of the same flavour.
Simple differential field extensions and effective bounds
James Freitag (University of California at Los Angeles, USA) and
Wei Li (Academy of Mathematics and Systems Science, China)
We establish several variations on Kolchin's differential primitive element theorem, and conjecture a generalization of Pogudin's primitive element theorem. These results are then applied to improve the bounds for the effective Differential Luroth theorem.