This paper deals with the notion of Gröbner δ-base for some rings of linear differential operators by adapting the works of W. Trinks, A. Assi, M. Insa and F. Pauer. We compare this notion with the one of Gröbner base for such rings. As an application we give some results on finiteness and on flatness of finitely generated left modules over these rings.
We consider the asymptotic growth of Hardy-field solutions of algebraic differential equations, e.g. solutions with no oscillatory component, and prove that no ‘sub-term’ occurring in a nested expansion of such can tend to zero more rapidly than a fixed rate depending on the order of the differential equation. We also consider series expansions. An example of the results obtained may be stated as follows.Let be an element of a Hardy field which has an asymptotic series expansion in , and ,...
Let be a commutative ring, a commutative -algebra and the filtered ring of -linear differential operators of . We prove that: (1) The graded ring admits a canonical embedding into the graded dual of the symmetric algebra of the module of differentials of over , which has a canonical divided power structure. (2) There is a canonical morphism from the divided power algebra of the module of -linear Hasse–Schmidt integrable derivations of to . (3) Morphisms and fit into a...
We prove that an irreducible polynomial derivation in positive characteristic is a Jacobian derivation if and only if there exists an (n-1)-element p-basis of its ring of constants. In the case of two variables we characterize these derivations in terms of their divergence and some nontrivial constants.
We study liftings or deformations of -modules ( 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 -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 -module in positive characteristic. At the end we compare the problems...
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 in the case of char k = p > 0.
Given a 3-dimensional vector field V with coordinates , and that are homogeneous polynomials in the ring k[x,y,z], we give a necessary and sufficient condition for the existence of a Liouvillian first integral of V which is homogeneous of degree 0. This condition is the existence of some 1-forms with coordinates in the ring k[x,y,z] enjoying precise properties; in particular, they have to be integrable in the sense of Pfaff and orthogonal to the vector field V. Thus, our theorem links the existence...
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.
We give a new proof of Miyanishi's theorem on the classification of the additive group scheme actions on the affine plane.
We prove that every locally nilpotent monomial k-derivation of k[X₁,...,Xₙ] is triangular, whenever k is a ring of characteristic zero. A method of testing monomial k-derivations for local nilpotency is also presented.
Dans cet article, nous étudions les modules libres de type fini sur l’anneau où est l’anneau des éléments analytiques dans une couronne de . D’une part, nous définissons, pour chaque nombre de , un rayon de convergence “générique" et nous montrons que celui-ci dépend continûment de . D’autre part, nous étudions l’existence et l’unicité d’un “antécédent de Frobenius".