The Kolchin Seminar in Differential Algebra

Saturday - Sunday November 23-24 2002

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Saturday Session

**William Sit
**

City College, CUNY

**Linear Differential Ideals, I**

2:00-3:00 PM

**Abstract:** This first part will be an introduction to the algebraic theory of systems of ordinary (or partial) linear homogeneous differential equations. As is well known, the set of solutions to such a system form a vector space over the constants. The special case when the dimension of this vector space is finite will be explored from several viewpoints. For the general case, when the dimension of the solution space is possibly infinite, there is associated with each system a canonical form and a field of constants.

**William Sit**** **

City College, CUNY

**Linear Differential Ideals, II**

3:30 - 4:30 PM

**Abstract:** The second part of the talk will be devoted to the geometric theory. The solution space, as an additive subgroup of affine n-space (n being the number of unknown functions), has a dimension polynomial (similar to the Hilbert polynomial) that provides certain invariants representing precise notions for the "arbitrariness" of the system. A relation among these invariants for the sum and intersection of two solution spaces exists and a similar relation continues to hold for solution sets of differential systems (which are not necessarily defined by linear homogeneous differential equations) if the solution sets are subgroups of matrix groups.

**Alexander Levin ****
**

The Catholic University of America

**Differential and Difference Dimension Polynomials**

5:00 - 6:00 PM

**Abstract:** The role of Hilbert polynomials in commutative algebra and algebraic geometry is well known. A similar role in differential algebra is played by the differential dimension polynomials introduced by E. Kolchin in 1964. In this talk we review basic facts about differential dimension polynomials and their difference analogs, discuss methods of computation of such polynomials, and consider their invariants. Then we sketch the proofs of theorems on multivariable dimension polynomials that generalize the classical results and give new invariants of differential and difference field extensions.

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Scientific Director of the CAFE Project (INRIA)

10:30 - 11:30 AM

**Abstract**: We say that a K(x)-basis W of the algebraic function field F = K[x,y]/(P(x,y)) is (globally) regular if its elements are integral over K[x] and if the differential system W' = A W is in Moser form at all the roots of disc(P). Any such basis can be used instead of the differential operator associated with P in order to compute topological properties of the curve, such as its genus or irreducible components. In addition, regular bases can replace the more expensive integral bases in various algorithms, such as the Hermite reduction for integrating algebraic functions. We also describe a simple algorithm for computing local or global regular bases and demonstrate its implementation in Maple, together with its application to computing the genus.

Manuel Bronstein

Scientific Director of the CAFE Project (INRIA)

**Special Function Solutions of y'' = r (x) y**

12:00 - 12:30 PM

**Abstract:** We describe an algorithm for computing special function solutions of the form
y = m(x) F(f(x)) of second order linear ordinary differential equations, where m is an unknown
Liouvillian function, f is an unknown rational function, and F is a known solution of a given second
order linear ordinary differential equation. The algorithm, which is based on finding an appropriate
point transformation between the equation defining F and the one to solve, is able to find all
rational transformations for a large class of functions F, in particular (but not only) the 0F1
and 1F1 special functions of mathematical physics, such as Airy, Bessel, Kummer and Whittaker
functions. It is also able to identify the values of the parameters entering those special functions,
and can be generalized to equations of higher order. We also demonstrate its implementation, which can
also be used online at
http://www-sop.inria.fr/cafe/Manuel.Bronstein/cathode/kovacic_demo.html

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**Coffee and tea will be served beginning at 1:30 PM on Saturday, and at 10:00 AM on Sunday Room HW706**

**Click here for directions to Hunter College and location of room**