F.J. Castro-Jiménez

Explicit Comparison Theorems for D -modules

We prove in an explicit way a duality formula between two A2-modulesMlog andMlog associated to a plane curve and we give an application of this duality to the comparison between Mlogand theA2 -module of rational functions along the curve. We treat the analytic case as well.

The analytic standard fan of a D-module

Abstract In this paper, we associate with any monogeneous module over the ring D of germs of linear differential operators at the origin of C n , with holomorphic coefficients, a combinatorial object which we call the standard fan of this D -module (see Section 6 for a precise geometric description of this object). The main tool of the proof is the homogenization technique and a convergent division theorem in the homogenization ring D [t]. This last result is the key tool to an extension to the analytic D -module case of our results in the algebraic case of the Weyl algebra (see Assi et al., J. Pure Appl. Algebra, 150 (1) (2000) 27–39.

The Gröbner fan of an An-module

Abstract Let I be a non-zero left ideal of the Weyl algebra A n of order n over a field k and let L: R 2n → R be a linear form defined by L(α,β)=∑ i=1 n e i α i +∑ i=1 n f i β i . If e i +f i ≥0 , then L defines a filtration F • L on A n . Let gr L (I) be the graded ideal associated with the filtration induced by F • L on I . Let finally U denote the set of all linear form L for which e i +f i ≥0 for all 1≤i≤n . The aim of this paper is to study, by using the theory of Grobner bases, the stability of gr L (I) when L varies in U . In a previous paper, we obtained finiteness results for some particular linear forms (used in order to study the regularity of a D -module along a smooth hypersurface). Here we generalize these results by adapting the theory of Grobner fan of Mora-Robbiano to the D -module case. Our main tool is the homogenization technique initiated in our previous paper, and recently clarified in a work by F. Castro-Jimenez and L. Narvaez-Macarro.

On the computation of Bernstein¿Sato ideals

Abstract In this paper we compare the approach of Briancon and Maisonobe for computing Bernstein–Sato ideals—based on computations in a Poincare–Birkhoff–Witt algebra—with the readily available method of Oaku and Takayama. We show that it can deal with interesting examples that have proved intractable so far.

Singularities of the hypergeometric system associated with a monomial curve

We compute, using D-module restrictions, the slopes of the irregular hypergeometric system associated with a monomial curve. We also study rational solutions and reducibility of such systems.

Testing the Logarithmic Comparison Theorem for Free Divisors

We propose in this work a computational criterion to test if a free divisor D ⊂ Cn verifies the Logarithmic Comparison Theorem (LCT); that is, whether the complex of logarithmic differential forms computes the cohomology of the complement of D in Cn . For Spencer free divisors D ≡ (f = 0), we solve a conjecture about the generators of the annihilating ideal of 1/f and make a conjecture on the nature of Euler homogeneous free divisors which verify LCT. In addition, we provide examples of free divisors defined by weighted homogeneous polynomials that are not locally quasi-homogeneous.

Logarithmic Comparison Theorem and some Euler homogeneous free divisors

Let D, x be a free divisor germ in a complex manifold X of dimension n > 2. It is an open problem to find out which are the properties required for D,x to satisfy the so-called Logarithmic Comparison Theorem (LCT), that is, when the complex of logarithmic differential forms computes the cohomology of the complement of D,x. We give a family of Euler homogeneous free divisors which, somewhat unexpectedly, does not satisfy the LCT.

LINEAR NESTED ARTIN APPROXIMATION THEOREM FOR ALGEBRAIC POWER SERIES

We give an elementary proof of the nested Artin approximation theorem for linear equations with algebraic power series coefficients. Moreover, for any Noetherian local subring of the ring of formal power series, we clarify the relationship between this theorem and the problem of the commutation of two operations for ideals: the operation of replacing an ideal by its completion and the operation of replacing an ideal by one of its elimination ideals. In particular we prove that a Grothendieck conjecture about morphisms of analytic/formal algebras and Artin’s question about linear nested approximation problem are equivalent.

Foreword by the Guest Editors

The papers in this Special Issue contain original research results on the rapidly growing area of Algorithmic Algebra and, more precisely, on Effective Methods in Rings of Differential Operators and some related rings. In particular, ideals in rings of differential operators are one of the most relevant sets of mathematical objects in this theory, since they play an analogous role to that of polynomial ideals in algebraic geometry. During recent years, many algorithms for manipulating differential operator ideals (and their induced D-modules), that rely heavily on Gröbner bases theory for differential operator rings, have been built up. In this Special Issue high-quality new papers on such topics have been collected, and we are convinced this volume would define a base-line for reference in the field. All contributions to this issue are framed in the field of effective methods in the theory of D-modules (i.e. the algebraic–geometric study of systems of linear partial differential equations) and related topics. Algorithms in D-modules are part of a larger area, namely effective differential algebra. In this sense, papers in this volume should be seen as complementary material to those published in previous issues of the Journal of Symbolic Computation such as the Special Issue on Symbolic Computation in Algebra, Analysis and Geometry (Guest Editors: Eduardo Cattani, Reinhard C. Laubenbacher), Journal of Symbolic Computation, Vol. 29, No. 4/5, May 2000 or Journal of Symbolic Computation, Vol. 28, No. 4/5, October/November 1999 (Guest Editors: William Sit, Manuel Bronstein).