**The
Kolchin Seminar in Differential Algebra**

**Saturday October 16**^{th}** 2004**

**Peter Landesman**

Graduate Center - CUNY
** Differential Polynomials with the Property that Each Differential
Zero Generates a Differential Field of Infinite Algebraic Transcendence Degree**

2:00-3:00PM

**Abstract:** Johnson, Reinhart and Rubel
have exhibited examples of differential polynomials with coefficients in a
differential field F such that each solution in a universal differential field
over F generates a field of infinite transcendence degree. This paper uses
their techniques to produce new examples.

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**Phyllis**** ****Cassidy**

**Parametrized Picard-Vessiot extensions (continued)**

**Abstract:** These talks describe a new research project, joint
work with Michael F. Singer. Consider the Kummer
equation

.

It is a second order
differential equation on the complex plane, in the operator , with
coefficients rational in , and, in a parameter . The equation has a regular singularity at , and, an irregular
singularity at . The special function , known as the lower incomplete gamma function, is a solution. Let
be a of .
If we adjoin and
all its and, derivatives to , we will have an example of a parametrized Picard-Vessiot
extension with Galois group the additive group . If, we then, adjoin , the Galois
group of the extension is the multiplicative group If we descend to ,
the differential extension field generated by has
Galois group the semi direct product , where is the subgroup of defined by the differential equation (equivalently, ).
We should note that, although has finite absolute
dimension (it depends on arbitrary constants), it can not be realized as an
algebraic subgroup of , for any
positive integer (it does not descend to
constants). is,
itself, the Galois group of the intermediate differential field over . The fact
that does not descend to
constants tells us that the parametrized Picard-Vessiot extension of is not a
classical Picard-Vessiot extension. It is, however, a classical Picard-Vessiot
extension of . It is time to step back, and, move from these interesting
examples to the general theory of parametrized Picard-Vessiot extensions. In
these talks, I show how the theory has a natural context: the ``new'' geometry
called *differential algebraic geometry*. Since the Galois groups are
linear, we remain comfortably in affine differential algebraic geometry. We
will also see how differential closures, developed by Shelah,
Kolchin, McGrail, and others, are efficient
replacements here for Kolchin's universal differential fields. We have three
sets of derivation operators to keep track of: The ``Galois theory'' derivation
, the set of
derivations , with respect to the parameters, and, the union of the two sets of derivations. The field of constants of , consisting of
functions of the parameters, is both a -field
and a -field. The interplay
of -closures and -closures
is very interesting. Parametrized Picard-Vessiot extensions are a special case
of Peter Landesman's new theory, *generalized strongly normal extensions*,
in which, the Galois groups are differential algebraic groups. Landesman's
theory is based on Kolchin's axiomatic treatment of differential algebraic
groups. Since the Galois groups of parametrized Picard-Vessiot extensions are
linear groups, the foundations of the theory can be developed easily *ab
initio*. Moreover, the theory is a striking example of the role that
differential algebraic groups play as symmetry groups of systems of polynomial
differential equations. The symmetry action here is the right regular
representation, which enables us to prove easily the Fundamental Theorem of
Galois theory, even in the case where the set of *isomorphisms*, rather
than the group of automorphisms, is given the structure of differential
algebraic group, and, no conditions are placed on the ground differential
field. This enables us to keep track of the dependence of the elements of the
extension field on the parameters, as well as on . In this generalization of Picard-Vessiot theory,
differential algebraic groups are realized as groups of deformations of systems
of differential equations that depend on parameters. The special function known
as the incomplete gamma function provides us with a realization of a
differential algebraic group of infinite absolute dimension (it depends on
infinitely many arbitrary constants) as a Galois group. Its derivative, , with respect
to , gives us an example of a parametrized Picard-Vessiot
extension of *finite *transcendence degree, which, nevertheless, is *not*
a classical Picard-Vessiot extension.

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**Phyllis
Cassidy****
**City College - CUNY

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**Coffee
and tea will be served beginning at 1:30 PM
Room HW706**

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