Kolchin Seminar in Differential Algebra
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Academic year 2005-2006.

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Saturday, October 22, 2005
Sally Morrison, Bucknell University, Lewisburg, PA
New methods in the computation of differential ideals

Computations in differential polynomial algebra inevitably lead one to consider ideals of the form  [P] : M\infty , where  P  is a finite set of differential polynomials and  M  is a multiplicative subset of the underlying differential polynomial ring. The work of Ritt and Kolchin exploits these ideals in the case that  P  is a coherent autoreduced set and  M  is the multiplicative subset generated by the initials and separants of the elements of  P. More recent computational techniques depend on the use of more general sets  P  and  M. In this talk we explore important properties of these more general ideals, and comment on the actual and potential use of these ideals in computational techniques.

For lecture notes click here.

Alexander Levin, Catholic University of America, Washington, DC
Differential dimension polynomials

The role of Hilbert polynomials in commutative algebra and algebraic geometry is well-known. Recall that the classical version of the existence theorem on Hilbert polynomial states that if  R[X1, ... ,X n is a naturally graded polynomial ring over a field  K  and  M = \bigoplusq Mq  is a finitely generated graded  R-module, then each component  Mq, considered as a vector  K-space, has finite dimension  \phiM(q)  and for sufficiently large  q, the function  \phiM(q)  is a polynomial in  q  whose degree is at most  n-1. This polynomial is called the Hilbert polynomial of the module  M.  A similar role in differential algebra is played by differential dimension polynomials introduced by E. Kolchin in 1964.

In this talk we discuss basic facts about differential dimension polynomials, outline methods of computation of such polynomials, and consider their invariants. We also discuss the connection between differential dimension polynomials and the concept of strength of a system of differential equations introduced by A. Einstein in his last works on Relativity.

The talk does not assume any knowledge beyond the standard undergraduate abstract algebra course.

For lecture notes click here.

Lourdes Juan, Texas Tech University, Lubbock, TX
Generic Picard-Vessiot extensions for connected-by-finite groups

Let  G be a linear algebraic group over the algebraically closed field  C. In this talk we will show how a generic Picard-Vessiot  G-extension can be produced when the group  G  is the semidirect product of its connected component  Go  by a finite group  H,  provided that the adjoint  H-action on  Lie(Go)  is faithful. The main ingredients are our previous construction of a generic extension with connected Galois group  G, a characterization of  H-equivariance based on the  C-linearity of the adjoint action obtained in this work, and a criterion of Mitschi-Singer to produce a Picard-Vessiot extension with group  H \ltimes Go.

For lecture notes click here.

Saturday, December 17, 2005
Alexey Ovchinnikov, North Carolina State University, Raleigh, NC
Tannakian categories for parametric differential equations

Tannaka's Theorem says that a linear algebraic group is determined by the category of vector spaces on which it acts. This has been used as a foundation of the Picard-Vessiot theory. We will discuss a similar result for linear differential algebraic groups and its connection with the Galois theory of parameterized linear differential equations.

Tobias Dyckerhoff, University of Pennsylvania, Philadelphia, PA
Picard-Vessiot theory over non-algebraically closed fields of constants

Differential Galois theory uses Picard-Vessiot extensions to describe the symmetries of linear differential equations. However, there are some difficulties when dealing with non-algebraically closed fields of constants. Galois descent provides a nice method to understand this situation. I will present an overview of this approach and describe relations to the theory of Tannakian categories.

For Diplomarbeit click here.

Jerald Kovacic, CCNY
Differential schemes

An affine differential scheme,  X = diffspec R, is similar to an affine scheme, except that we start with a differential ring  R  and consider differential prime ideals. There is a canonical mapping of  R  into the ring of global section of  X.  In scheme theory this mapping is an isomorphism, not so for differential schemes. We can easily determine the kernel. It is the differential ideal of differential zeros. Surjectivity is missing because of the existence of differential units and the lack of a common denominator. We shall also discuss other "challenges", such as the existence of products. For differential group schemes we have the challenge that they need not be linear,  R  need not be a differential Hopf algebra. This is an elementary talk. We assume the audience knows the definition of spec but little else.

For lecture notes click here.

Saturday, March 18, 2006

Hans Schoutens, New York City College of Technology, CUNY
Russell Miller, Queens College, CUNY
Hrushovski's proof of the Mordell-Lang Conjecture
Part I: Jet-spaces (Hans Schoutens)
Part II: Model-theoretic differential algebra (Russell Miller)

The Mordell-Lang Conjecture, stated below and presented in [1], does not involve any differential operators. Nevertheless, differential algebra plays a substantial role in Hrushovski's proof of this result, at least in the case of characteristic 0.

Mordell-Lang Conjecture (relative version, for function fields): Let  k0 be two distinct algebraically closed fields. Let  X  be an infinite subvariety of an abelian variety  A , both defined over  K , and let  Γ  be a subgroup of  A(K)  of finite rank. Also suppose that  X(K) Γ  is Zariski-dense in  X(K) . Then either

(1)  X(K) Γ  is a finite union of cosets of subgroups of  Γ , or
(2) the data 'descend to  k0 ', in the sense that there exist a subabelian variety  B  of  A , an abelian variety  B0  and a subvariety  X0  of  B0  both defined over  k0 , and a bijective morphism  h  from  B  onto  B0 , such that  X = a + h-1(X0 for some  a.

Part I (jet-spaces):

After Hrushovski's original proof was published, Pillay and Ziegler [3] found a way around part of the heavy-duty theory (Zariski geometries) that was needed in the proof by introducing the differential analogue of a jet-space. Model-theory is now only present in the background of this part of the proof.

Part II (differential algebra):

We introduce the model theory of the situation and investigate the reasons why derivations are useful in this context, and what one can do with differential algebra that could not have been done (at least, not so easily) with ordinary algebraic geometry.


[1] Bouscaren, E., ed., Model Theory and Algebraic Geometry: An introduction to E. Hrushovski's proof of the geometric Mordell-Lang conjecture. (Berlin: Springer, 1999.)

[2] Marker, D., Model Theory of Differential Fields. (Berlin: Springer, 1996.)

[3] Pillay, A. and Ziegler, M., Jet spaces of varieties over differential and difference fields. Selecta Math. 9 (2003), no. 4, 579-599.

Oleg Golubitsky, School of Computing, Queen's University, Kingston, Ontario
A bound for the orders of derivatives in the Rosenfeld-Gröbner algorithm

(in collaboration with M. Kondratieva, M. Moreno Maza, and A. Ovchinnikov)

We consider the Rosenfeld-Gröbner algorithm for computing a regular decomposition of a radical differential ideal generated by a set of ordinary differential polynomials in  n  indeterminates. For a set of ordinary differential polynomials  F , let  M(F)  be the sum of maximal orders of differential indeterminates occurring in  F . We propose a modification of the Rosenfeld-Gröbner algorithm, in which for every intermediate polynomial system  F , the bound  M(F) ≤ (n-1)!M(F0 holds, where  F0  is the initial set of generators of the radical differential ideal.

In particular, the resulting regular systems satisfy the bound. Since regular ideals can be decomposed into characterizable components algebraically, the bound also holds for the orders of derivatives occurring in the characteristic decomposition of a radical differential ideal.

For lecture notes click here.

Saturday, May 6, 2006
1:30 - 2:30 PM
Xiao-Shan Gao, Institute of Systems Sciences, Chinese Academy of Sciences

Difference characteristic set and resolvent

In this talk, we will give an introduction to our current work on computational difference algebra via the characteristic set method. We first prove several basic properties for difference ascending chains including a necessary and sufficient condition for an ascending chain to be the characteristic set of its saturation ideal and a necessary and sufficient condition for an ascending chain to be the characteristic set of a reflexive prime ideal. We then propose an algorithm to decompose the zero set of a finite set of difference polynomials into the union of zero sets of certain ascending chains. As a consequence of the zero decomposition, we give a new algorithm for the perfect ideal membership problem. Finally, we introduce a new theory of resolvent systems for certain difference ideals and introduce algorithms to compute them.

For lecture notes click here.

2:50 - 3:50 PM
Lourdes Juan, Texas Tech University, Lubbock
On the structure of Picard-Vessiot extensions

(joint work with Arne Ledet)

Let  k  be a differential field of characteristic zero with algebraically closed field of constants. A Picard-Vessiot extension  K is the function field of a  k  irreducible  G  torsor, where  G  is the differential Galois group of the extension. In this talk we will show how to construct examples of extensions which are the function field of non-trivial  G  torsors and that, when a good description of the torsors is available, generic extensions for  G can be produced.

For lecture notes click here , see also the web site

4:10 - 5:10 PM
Julia Hartmann, University Of Pennsylvania, Philadelphia

Inverse problems in differential Galois theory
5:30 - 6:30 PM
Yang Zhang, Brandon University, Brandon, Canada

Non-commutative existence and uniqueness theory for partial differential equations

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