Unipotent differential algebraic groups as parameterized differential Galois groups

Abstract

We deal with aspects of the 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.

Parallel telescoping and parameterized Picard-Vessiot theory

“…They show that the additive group (F, +) cannot be the Galois group of a parameterized Picard-Vessiot extension of this field. In rest of this section, we show a similar result for fields of rational functions in several variables (for other results concerning the inverse problem, see [10,23,24,25,28]). The key tool will be the fact that parallel telescopers always exist for compatible rational functions.…”

“…Algorithms for first and second order parameterized equations over F (x), where n = 1, appear in [2,9]. An algorithm to determine if a parameterized equation of arbitrary order has a unipotent PPV-group (or even certain kinds of extensions of such a group) as well as an algorithm to compute the group appears in [24]. An algorithm to determine if a parameterized equation has a reductive PPV-group and compute it if it does appears in [23].…”

Computing the differential Galois group of a parameterized second-order linear differential equation

“…Building on this work, unconditional algorithms to compute G are given in [1] in the setting of one parametric derivation, and in [3] for several parametric derivations but still assuming that r1 = 0. Algorithms for higher-order equations are developed in [17,18]. After performing a change of variables on (2), we obtain an associated equation (3) of the form δ 2…”

“…See [17,18] for a general discussion of unipotent radicals and reductivity in the context of LDAGs, and for algorithms to compute PPV-groups for higher-order equations which either are reductive, or whose quotient by the unipotent radical is differentially constant.…”

On the parameterized differential inverse Galois problem over k((t))(x)

“…This implies in particular that every semisimple linear algebraic group defined over U is a parameterized differential Galois group over U (x). For unipotent and reductive linear differential algebraic groups, there are also recent results that give characterizations which groups are finitely generated as differential algebraic groups [9,8].…”