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Unipotent differential algebraic groups as parameterized differential Galois groups

Part of: Other groups of matrices Differential algebra Differential and difference algebra Linear algebraic groups and related topics Differential equations in the complex domain

Published online by Cambridge University Press:  18 July 2013

Andrey Minchenko ,
Alexey Ovchinnikov  and
Michael F. Singer
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)
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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 $0$.

Keywords

differential algebraic groupsparameterized differential Galois theoryalgorithms

MSC classification

Secondary: 12H05: Differential algebra 12H20: Abstract differential equations 13N10: Rings of differential operators and their modules 20G05: Representation theory 20H20: Other matrix groups over fields 34M15: Algebraic aspects (differential-algebraic, hypertranscendence, group-theoretical)
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 13 , Issue 4 , October 2014 , pp. 671 - 700
Copyright
©Cambridge University Press 2013 

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