Unipotent differential algebraic groups as parameterized differential Galois groups. (October 2014) - Minchenko, Andrey; Ovchinnikov, Alexey; Singer, Michael F.
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) <alternatives><inline-graphic xlink:href="ark:/27927/pgh1239s3m8" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$G$]]> </alternatives> is a PPV Galois group over these fields if and only if <alternatives><inline-graphic xlink:href="ark:/27927/pgh1239s3fh" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$G$]]> </alternatives> contains a Kolchin-dense finitely generated group. We show that, for a class of LDAGs <alternatives><inline-graphic xlink:href="ark:/27927/pgh1239s3s1" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$G$]]> </alternatives>, including unipotent groups, <alternatives><inline-graphic xlink:href="ark:/27927/pgh1239s4dg" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$G$]]> </alternatives> is such a group if and only if it has differential type <alternatives><inline-graphic xlink:href="ark:/27927/pgh1239s488" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$0$]]> </alternatives>. 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 <alternatives><inline-graphic xlink:href="ark:/27927/pgh1239s399" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$0$]]> </alternatives>.