PRINCIPAL DIFFERENTIAL IDEALS AND A GENERIC INVERSE DIFFERENTIAL GALOIS PROBLEM FOR GL_n

ABSTRACT

We characterize the principal differential ideals of a polynomial ring in indeterminates with coefficients in the ring of differential polynomials in indeterminates and derivation given by a “general” element of and use this characterization to construct a generic Picard-Vessiot extension for . In the case when the differential base field has finite transcendence degree over its field of constants we provide necessary and sufficient conditions for solving the inverse differential Galois problem for these groups via specialization from our generic extension.

ACKNOWLEDGMENT

This paper was written while the author held an NSF-funded postdoctoral fellowship at the MSRI. The author wishes to thank the MSRI for its hospitality.