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Lifting D -modules from positive to zero characteristic

João Pedro P. dos Santos (2011)

Bulletin de la Société Mathématique de France

We study liftings or deformations of D -modules ( D is the ring of differential operators from EGA IV) from positive characteristic to characteristic zero using ideas of Matzat and Berthelot’s theory of arithmetic D -modules. We pay special attention to the growth of the differential Galois group of the liftings. We also apply formal deformation theory (following Schlessinger and Mazur) to analyze the space of all liftings of a given D -module in positive characteristic. At the end we compare the problems...

Linear derivations with rings of constants generated by linear forms

Piotr Jędrzejewicz (2008)

Colloquium Mathematicae

Let k be a field. We describe all linear derivations d of the polynomial algebra k[x₁,...,xₘ] such that the algebra of constants with respect to d is generated by linear forms: (a) over k in the case of char k = 0, (b) over k [ x p , . . . , x p ] in the case of char k = p > 0.

Linear differential equations and Hurwitz series

William F. Keigher, V. Ravi Srinivasan (2011)

Banach Center Publications

In this article, we study solutions of linear differential equations using Hurwitz series. We first obtain explicit recursive expressions for solutions of such equations and study the group of differential automorphisms of the solutions. Moreover, we give explicit formulas that compute the group of differential automorphisms. We require neither that the underlying field be algebraically closed nor that the characteristic of the field be zero.

Liouvillian first integrals of differential equations

Guy Casale (2011)

Banach Center Publications

In this paper we generalize to any dimension and codimension some theorems about existence of Liouvillian solutions or first integrals proved by M. Singer in Liouvillian first integrals of differential equations (1992) for first order differential equations.

Local derivations for quotient and factor algebras of polynomials

Andrzej Nowicki, Ilona Nowosad (2003)

Colloquium Mathematicae

We describe all Kadison algebras of the form S - 1 k [ t ] , where k is an algebraically closed field and S is a multiplicative subset of k[t]. We also describe all Kadison algebras of the form k[t]/I, where k is a field of characteristic zero.

Local derivations in polynomial and power series rings

Janusz Zieliński (2002)

Colloquium Mathematicae

We give a description of all local derivations (in the Kadison sense) in the polynomial ring in one variable in characteristic two. Moreover, we describe all local derivations in the power series ring in one variable in any characteristic.

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