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.

minor revision

References in corpus (3)

• Parameterized Picard-Vessiot extensions and Atiyah extensions
• Tannakian categories, linear differential algebraic groups, and parameterized linear differential equations
• Tannakian approach to linear differential algebraic groups

Cited by in corpus (14)

• Parameterized Picard-Vessiot extensions and Atiyah extensions
• Reductive linear differential algebraic groups and the Galois groups of parameterized linear differential equations
• Isomonodromic differential equations and differential categories
• Calculating differential Galois groups of parametrized differential equations, with applications to hypertranscendence
• Computation of the difference-differential Galois group and differential relations among solutions for a second-order linear difference equation
• Difference algebraic relations among solutions of linear differential equations
• Computation of the unipotent radical of the differential Galois group for a parameterized second-order linear differential equation
• Finiteness properties of affine difference algebraic groups
• Computing differential Galois groups of second-order linear $q$-difference equations
• Difference integrability conditions for parameterized linear difference and differential equations
• Tannakian categories with semigroup actions
• Parameterized Differential Equations over k((t))(x)
• Calculating Galois groups of third order linear differential equations with parameters
• Some applications of the parameterized Picard-Vessiot theory