Unipotent differential algebraic groups as parameterized differential Galois groups. (October 2014)

• Record Type: Journal Article
• Authors: Minchenko, Andrey; Ovchinnikov, Alexey; Singer, Michael F.
• Is Part Of: Journal of the Institute of Mathematics of Jussieu. Volume 13:Number 4(2014:Oct.)
• Page Start: 671
• Page End: 700
• Publication Date: 2014-10
• Subjects: Mathematics -- Periodicals; 510.5
• DOI: 10.1017/S1474748013000200
• Languages: English
• ISSNs: 1474-7480

Abstract

<abstract abstract-type="normal"> <title>Abstract</title> <p>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) <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh1239s3m8" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$G$]]></tex-math></alternatives></inline-formula> is a PPV Galois group over these fields if and only if <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh1239s3fh" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$G$]]></tex-math></alternatives></inline-formula> contains a Kolchin-dense finitely generated group. We show that, for a class of LDAGs <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh1239s3s1" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$G$]]></tex-math></alternatives></inline-formula>, including unipotent groups, <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh1239s4dg" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$G$]]></tex-math></alternatives></inline-formula> is such a group if and only if it has differential type <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh1239s488" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$0$]]></tex-math></alternatives></inline-formula>. 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 <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh1239s399" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$0$]]></tex-math></alternatives></inline-formula>.</p> </abstract>