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.

******************************************

Phyllis Cassidy
City College - CUNY
Parametrized Picard-Vessiot extensions (continued)
3:30 - 4:30PM

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

$\displaystyle \frac{d^{2}y}{dx^{2}}-\left( \frac{t-x-1}{x}\right) \frac{dy}{dx}=0$ . $\displaystyle %%$

It is a second order differential equation on the complex plane, in the operator $\frac{d}{dx}$ , with coefficients rational in , and, in a parameter . The equation has a regular singularity at , and, an irregular singularity at $x=\infty$ . The special function $\gamma\left( t,x\right) =\int\limits_{0}^{x}s^{t-1}e^{-s}ds$ , known as the lower incomplete gamma function, is a solution. Let $\mathcal{C}$ be a $\partial_{t}-closure$ of $\mathbb{Q}\left( t\right)$ . If we adjoin $\gamma\left( t,x\right)$ and all its $\partial_{x}-$ and, $\partial_{t}-$ derivatives to $\mathcal{F}=\mathcal{C}\left( x,\log x,x^{t-1}e^{-x}\right)$ , we will have an example of a parametrized Picard-Vessiot extension with Galois group the additive group $G_{a}\left( \mathcal{C}\right)$ . If, we then, adjoin $e^{\gamma(t,x)}$ , the Galois group of the extension is the multiplicative group $G_{m}\left( \mathcal{C}\right)$ If we descend to $\mathcal{C}\left( x\right)$ , the differential extension field generated by $\gamma\left( t,x\right)$ has Galois group the semi direct product $G_{a}\left( \mathcal{C}\right) \cdot G$ , where is the subgroup of $G_{m}\left( \mathcal{C}\right)$ defined by the differential equation $y\partial_{t}^{2}y-\left( \partial_{t}y\right) ^{2}=0$ (equivalently, $\partial_{t}^{2}\log y=0$ ). We should note that, although has finite absolute dimension (it depends on arbitrary constants), it can not be realized as an algebraic subgroup of $GL(n,\mathcal{C})$ , for any positive integer (it does not descend to constants). is, itself, the Galois group of the intermediate differential field $\mathcal{C}\left( x,\log x,x^{t-1}e^{-x}\right)$ over $\mathcal{C}(x)$ . The fact that does not descend to constants tells us that the parametrized Picard-Vessiot extension $\mathcal{C}\left( x,\log x,x^{t-1}e^{-x}\right)$ of $\mathcal{C}(x)$ is not a classical Picard-Vessiot extension. It is, however, a classical Picard-Vessiot extension of $\mathcal{C}\left( x,\log x\right)$ . 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 $\partial_{x}$ , the set $\Delta \acute{}%%$ of derivations $\partial_{t_{1}},...,\partial_{t_{m}}$ , with respect to the parameters, and, the union $\Delta$ of the two sets of derivations. The field of constants of $\partial_{x}$ , consisting of functions of the parameters, is both a $\Delta \acute{}%%$ -field and a $\Delta$ -field. The interplay of $\Delta$ -closures and $\Delta \acute{}%%$ -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, $x^{t-1}e^{-x}$ , 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.