MACIS 2015 Session (SS8): Computational theory of differential and difference equations

Organizers

• (RWTH Aachen University, Germany)
• (CUNY Queens College and Graduate Center, USA)
• (University of Pennsylvania, USA)

Aim and Scope

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

Publications

• 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, published by Birkhauser/Springer, will be organized after the conference by session organizers. REGULAR (not SHORT) papers would be considered for these special issues.

Submission Guidelines

• If you would like to give a talk at MACIS, you need to submit at least a SHORT paper -- see guideline 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.

Talks/Abstracts

• Dimension Polynomials of Intermediate Fields of Inversive Difference Field Extensions

  Alexander Levin (The Catholic University of America, USA)
  
  Abstract: 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)
  
  Abstract: 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)
  
  Abstract: 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)
  
  Abstract: 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.