Géométrie des points épais
Gröbner δ-bases and Gröbner bases for differential operators
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.
Growth orders occurring in expansions of Hardy-field solutions of algebraic differential equations
Growth orders occurring in expansions of Hardy-field solutions of algebraic differential equations
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 ,...
Hasse–Schmidt derivations, divided powers and differential smoothness
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...
Heinrich's counterexample to Azevedo's conjecture.
Idéaux différentiels
Integral differentials and Fermat congruences.
Integration of differential forms on schemes.
Irreducible Jacobian derivations in positive characteristic
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.
Jacobian criteria for complete intersections. The graded case.
Kähler differentials of affine monomials curves.
Lectures on minimal models
Lifting -modules from positive to zero characteristic
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...
Linear derivations with rings of constants generated by linear forms
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.
Liouvillian first integrals of homogeneouspolynomial 3-dimensional vector fields
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...
Local derivations in polynomial and power series rings
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.
Locally finite iterative higher derivations on k[x,y]
We give a new proof of Miyanishi's theorem on the classification of the additive group scheme actions on the affine plane.
Locally Nilpotent Monomial Derivations
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.
Modules des differentielles en caracteristique p.