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Unipotent differential algebraic groups as parameterized differential Galois groups

Published online by Cambridge University Press:  18 July 2013

Andrey Minchenko
Affiliation:
The Hebrew University of Jerusalem, Einstein Institute of Mathematics, Jerusalem, 91904, Israel (an.minchenko@gmail.com)
Alexey Ovchinnikov
Affiliation:
CUNY Queens College, Department of Mathematics, 65-30 Kissena Blvd, Queens, NY 11367, USA CUNY Graduate Center, Department of Mathematics, 365 Fifth Avenue, NY 10016, USA (aovchinnikov@qc.cuny.edu)
Michael F. Singer
Affiliation:
North Carolina State University, Department of Mathematics, Raleigh, NC 27695-8205, USA (singer@ncsu.edu)

Abstract

We deal with aspects of direct and inverse problems in parameterized Picard–Vessiot (PPV) theory. It is known that, for certain fields, a linear differential algebraic group (LDAG) $G$ is a PPV Galois group over these fields if and only if $G$ contains a Kolchin-dense finitely generated group. We show that, for a class of LDAGs $G$, including unipotent groups, $G$ is such a group if and only if it has differential type $0$. We give a procedure to determine if a parameterized linear differential equation has a PPV Galois group in this class and show how one can calculate the PPV Galois group of a parameterized linear differential equation if its Galois group has differential type $0$.

Type
Research Article
Copyright
©Cambridge University Press 2013 

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