Continuous derivations

Morrison, Sally

doi:10.1016/0021-8693(87)90058-5

Cited by 4 publications

(3 citation statements)

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“…Let us remark also that the present work introduces a new algebraic perspective in the study of singularities of plane and higher dimensional foliations (see [2]), which up to now have been studied mainly with complex analytic tools-either complex analysis or resolution of singularities. From the point of view of differential algebra, the results presented here give an alternative approach to Kolchin-Morrison's theory of continuous derivations [3], [4].…”

Section: Introductionmentioning

confidence: 99%

The Valuative Theory of Foliations

Ayuso¹

2002

Can. j. math.

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Abstract. This paper gives a characterization of valuations that follow the singular infinitely near points of plane vector fields, using the notion of L'Hôpital valuation, which generalizes a well known classical condition. With that tool, we give a valuative description of vector fields with infinite solutions, singularities with rational quotient of eigenvalues in its linear part, and polynomial vector fields with transcendental solutions, among other results.

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Section: Introductionmentioning

confidence: 99%

The Valuative Theory of Foliations

Ayuso¹

2002

Can. j. math.

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“…Cette notion de continuité de la dérivation par rapport à la topologie ν-adique généralise celle de « valuation différentielle » rencontrée dans [9,7,8,5,4,10]. Dans la lignée du Théorème 2.3, on obtient : Puisque l'on doit conserver la borne, il suffit de démontrer l'énoncé dans le cadre des extensions finies.…”

Section: Valuations Invariantes Et Continuité Des Dérivationsunclassified

Valuations invariantes pour l'action des groupes de Galois différentiels

Duval¹

2004

Comptes Rendus. Mathématique

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“…In this case, the degree deg(D) of any k-derivation D is finite. The boundedness of the degree for a k-derivation is analogous to the notion of continuity of a derivation with respect to a valuation (see [7]). Definition 1.4.…”

Section: Introductionmentioning

confidence: 98%

Invariant hypersurfaces for derivations in positive characteristic

Bonnet

2007

Journal of Pure and Applied Algebra

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Let A be an integral k-algebra of finite type over an algebraically closed field k of characteristic p > 0. Given a collection D of k-derivations on A, that we interpret as algebraic vector fields on X = Spec(A), we study the group spanned by the hypersurfaces V ( f ) of X invariant under D modulo the rational first integrals of D. We prove that this group is always a finite dimensional F p -vector space, and we give an estimate for its dimension. This is to be related to the results of Jouanolou and others on the number of hypersurfaces invariant under a foliation of codimension 1. As a application, given a k-algebra B between A p and A, we show that the kernel of the pull-back morphism Pic(B) → Pic(A) is a finite F p -vector space. In particular, if A is a UFD, then the Picard group of B is finite.

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Continuous derivations

Cited by 4 publications

References 3 publications

The Valuative Theory of Foliations

The Valuative Theory of Foliations

Valuations invariantes pour l'action des groupes de Galois différentiels

Invariant hypersurfaces for derivations in positive characteristic

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