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Journals and Conferences
We present an algorithm that can be used to check whether a given derivation of the complex affine plane has an algebraic solution and discuss the performance of its implementation in the computer algebra system Singular.
A more recent milestone in the subject was J. P. Jouanolou's lecture notes . An often quoted result from these notes states that the set of holomorphic foliations of the complex projective plane P that do not have an algebraic solution is dense in the space that parametrizes the foliations; see [Jouanolou 1979, Chapter 4, p. 157 ff.]. In order to… (More)
Commutative differentially simple rings have proved to be quite useful as a source of examples in non-commutative algebra. In this paper we use the theory of holomorphic foliations to construct new families of derivations with respect to which the polynomial ring over a field of characteristic zero is differentially simple.
We classify simple derivations induced by unimodular rows of length 2 whose entries have degree 2, over the ring of complex polynomials in two variables. As part of the proof we give several new families of simple derivations over this ring.
The most important geometric invariant in the theory of D-modules is the characteristic variety. Let X be a smooth complex algebraic variety and denote by D its sheaf of rings of differential operators. If M is a X Ž . coherent sheaf of modules over D then its characteristic variety Ch M X is a subvariety of the cotangent bundle T X. The commutator of… (More)
We give new examples of affine sufaces whose rings of coordinates are d-simple and use these examples to construct simple nonholonomic Dmodules over these surfaces.
— We present a constructive proof of the fact that the set of algebraic Pfaff equations without algebraic solutions over the complex projective plane is dense in the set of all algebraic Pfaff equations of a given degree. Résumé. — Nous présentons une preuve constructive du fait que l’ensemble des équations de Pfaff sans solutions algébriques sur le plan… (More)
We propose an algorithm that uses Gröbner bases to compute the resolution of the singularities of a foliation of the complex projective plane.