Related papers: Existence of $\partial$-parameterized Picard-Vessi…
We demonstrate existence and uniqueness of Picard--Vessiot extensions satisfying prescribed properties, for systems of linear differential equations over a field satisfying the same properties, under some closure assumptions on the field of…
We prove the existence of real Picard-Vessiot extensions for real partial differential fields with real closed field of constants. We establish a Galois correspondence theorem for these Picard-Vessiot extensions and characterize real…
The existence of a Picard-Vessiot extension for a homogeneous linear differential equation has been established when the differential field over which the equation is defined has an algebraically closed field of constants. In this paper, we…
In this paper, we establish Galois theory for partial differential systems defined over formally real differential fields with a real closed field of constants and over formally $p$-adic differential fields with a $p$-adically closed field…
Let F be a differential field with field of constants C. We assume C to be algebraically closed and of characteristic 0. The complete Picard--Vessiot closure of F is a differential field extension of F with the same constants C as F, which…
We prove some existence results on parameterized strongly normal extensions for logarithmic equations. We generalize a result in [Wibmer, Existence of d-parameterized Picard-Vessiot extensions over fields with algebraically closed…
Generalizing Atiyah extensions, we introduce and study differential abelian tensor categories over differential rings. By a differential ring, we mean a commutative ring with an action of a Lie ring by derivations. In particular, these…
This expository paper presents some applications of the parameterized Picard-Vessiot theory for ordinary differential equations, mainly related to monodromy.
Let F be a differential field of characteristic zero. In this article, we construct Picard-Vessiot extensions of F whose differential Galois group is isomorphic to the full unipotent subgroup of the upper triangular group defined over the…
For a linear differential equation defined over a formally real differential field K with real closed field of constants k, Crespo, Hajto and van der Put proved that there exists a unique formally real Picard- Vessiot extension up to…
We prove that if T is a theory of large, bounded, fields of characteristic zero, with almost quantifier elimination, and T_D is the model companion of T + "D is a derivation", then for any model U of T_D, and differential subfield K of U…
We present a characterization of differential Picard-Vessiot extensions which is analogous to the field theoretic characterization of Galois extensions in classical Galois theory.
We consider differential modules over real and p-adic differential fields such that their field of constants is real closed (respectively p-adically closed). Using Deligne's work on Tannakian categories and a result of Serre on Galois…
In this note, we describe a method to construct the Picard-Vessiot ring of a given linear differential equation.
Let K be differential field with algebraically closed field of constants. Let K^diff be a differential closure of K, and L the (iterated) Picard-Vessiot closure of K inside K^diff. Let G be a linear differential algebraic group over K and X…
In this paper, we generalize the definition of the differential Galois group and the Galois correspondence theorem established previously for Picard-Vessiot extensions of real differential fields with real closed field of constants to any…
For a differential field $F$ having an algebraically closed field of constants, we analyze the structure of Picard-Vessiot extensions of $F$ whose differential Galois groups are unipotent algebraic groups and apply these results to study…
Assuming that the differential field $(K,\delta)$ is differentially large, in the sense of Le\'on S\'anchez and Tressl, and "bounded" as a field, we prove that for any linear differential algebraic group $G$ over $K$, the differential…
In this paper, we prove a new characterization theorem for Picard-Vessiot extensions whose differential Galois groups have solvable identity components.
We compare several definitions of the Galois group of a linear difference equation that have arisen in algebra, analysis and model theory and show, that these groups are isomorphic over suitable fields. In addition, we study properties of…