Kolchin Seminar in Differential Algebra
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Academic year 2006-2007.

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Saturday, November 18, 2006, 2PM
Oleg Golubitsky, Ontario Research Centre for Computer Algebra and University of Western Ontario

Canonical representation and extension of rankings

A ranking is a total order relation on the Cartesian product of an abelian group and a finite set which is compatible with the group operation. The cases when the abelian group is that of m-tuples of integer, rational, or real numbers are of particular interest, since they are directly related to the problem of classification of rankings on partial derivatives, and the latter are essential for elimination algorithms in differential algebra.

When the finite set in the above Cartesian product consists of one element, rankings become group orders. Every order on the group of integer m-tuples extends to an order on rational m-tuples, which in turn extends to an order on real m-tuples. This chain of extensions gives a complete characterization of orders on integer m-tuples in terms of real matrices and, consequently, classifies admissible monomial orderings.

G. Carra Ferro and W. Sit characterized general rankings as transitive families of cuts of the abelian group; when the group is that of integer m-tuples, they obtained a representation for the cuts by real matrices. However, the problem of checking the transitivity condition for a given family of cuts is open. The existence of the above chain of extensions from integer to rational and real m-tuples for rankings is also an open problem. Finally, the problem of constructing a canonical representation of rankings is open.

We address the above problems by extending the results of C. Rust and G. Reid, which allow to represent rankings by finite trees of Riquier pre-rankings. Time permitting, we will also discuss an algorithm that determines whether a given finite relation on the Cartesian product of the group of integer m-tuples and a finite set is contained in a ranking.

Alexander Levin, The Catholic University of America, Washington, DC

Gröbner bases with respect to several term orderings and multivariate dimension polynomials

Let  D = K[X]  be a ring of Ore polynomials over a field  K  and let a partition of the set of indeterminates  X  into i> p  disjoint subsets be fixed. We introduce a special type of reduction in a free  D-module and develop the corresponding Gröbner basis technique that allows one to prove the existence of a Hilbert-type dimension polynomial in  p  variables associated with a finitely generated filtered  D-module. We also outline methods of computation and determine invariants of such a polynomial. As an application, we obtain generalizations of classical theorems on dimension polynomials of differential and difference field extensions and determine new invariants of such extensions. We will also discuss some other applications of the developed technique.

Alexander A. Mikhalev, Moscow University and City College

Free differential calculus for free algebras and differential Lie superalgebras

For an algebra of a homogeneous variety of linear algebras we consider the universal multiplicative enveloping algebra (we follow the monograph "Structure and Representations of Jordan Algebras", Amer. Math. Soc., 1968, by N. Jacobson). A variety of algebras is said to be Schreier if any subalgebra of this variety is a free algebra of the same variety of algebras. The main types of Schreier varieties of algebras over a field are the variety of all algebras, the variety of all commutative algebras, the variety of all anticommutative algebras, the variety of all Lie algebras, the varieties of all color Lie superalgebras, the varieties of all Lie p-algebras and of all color Lie p-superalgebras (if the main field has prime characteristic). For a free algebra of a Schreier variety of algebras we consider its multiplicative enveloping algebra which is a free associative algebra. Using the fact that a free associative algebra is a free ideal ring it gives a possibility to define universal derivation of free algebras (partial derivatives are components of the action of the universal derivation).

We study orbits of elements of free algebras of Schreier varieties of algebras under the action of the automorphism group. A set of elements of a free algebra is said to be primitive if it is a subset of some set of free generators of this algebra. Using free differential calculus matrix criteria for a set of elements of a free algebra of a Schreier variety of algebras to be primitive are obtained. It gives a possibility to construct fast algorithms of computer algebra to recognize primitive systems of elements. We construct also an algorithm to compute a rank of a system of elements of a free algebra and prove that endomorphisms preserving primitivity of elements are automorphisms.

Differential Lie algebras appeared in papers of R.Baer (1927) and N.Jacobson (1937) devoted to derivations of associative rings. V.K.Kharchenko used them (1979) to study differential identities of semiprime rings. General theory of differential Lie algebras was exposed in the monographs

K.I.Beidar, W.Martindale, and A.V.Mikhalev, Rings with Generalized Identities. Marcel Dekker, 1996.
V.K.Kharchenko, Derivations of Associative Rings. Kluwer, 1991.

We consider differential Lie superalgebras and prove analogs of Poincare-Birkhoff-Witt theorem for universal enveloping algebras of these algebras. In the proof we use the theory of Groebner-Shirshov bases of ideals of free algebras.

Saturday, December 16, 2006, 2PM
Li Guo, Rutgers University at Newark
Differential algebraic Birkhoff decomposition and renormaliztion of multiple zeta values

The algebraic approach of Connes and Keimer on renormalizaion of quantum field theory views the process as a special case of an algebraic Birkhoff decomposition. We give a differential algebra version of this decomposition and show how this can be applied to the study of multiple zeta values.

Evelyne Hubert, INRIA Sophia-Antipolis and IMA Minneapolis
Rational and algebraic invariants of a group action

We consider a rational group action on the affine space and propose a construction of a finite set of rational invariants and a simple algorithm to rewrite any rational invariant in terms of those generators.

We introduce a finte set of replacement invariants that are algebraic functions of the rational invariants. They are the algebraic analogues of the Cartan's normalized invariants and give rise to a trivial rewriting.

The construction can be show to be an algebraic analogue of the moving frame construction of local invariants [Fels and Olver 1999] and can be applied to compute differential invariants.

This is joint work with Irina Kogan, North Carolina State University.

For lecture notes click here.

Jacques-Arthur Weil, Université de Limoges
Galoisian methods for studying integrability of dynamical systems (generally hamiltonian), theory and practice

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