Computational methods in algebra and analysis

This paper describes some applications of Computer Algebra to Algebraic Analysis also known as D-module theory, i.e. the algebraic study of the systems of linear partial differential equations. Gröbner bases for rings of linear differential operators are the main tools in the field. We start by giving a short review of the problem of solving systems of polynomial equations by symbolic methods. These problems motivate some of the later developed subjects.Ministerio de Ciencia y TecnologíaJunta de Andalucí

On the computation of Bernstein–Sato ideals

In this paper we compare the approach of Brianc¸onand Maisonobe for computing Bernstein–Sato ideals—based on computations in a Poincar´e–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.Ministerio de Ciencia y Tecnología BFM-2001-3164Junta de Andalucía FQM-33

Explicit Comparison Theorems for D -modules

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

Gevrey expansions of hypergeometric integrals I

We study integral representations of the Gevrey series solutions of irregular hypergeometric systems. In this paper we consider the case of the systems associated with a one row matrix, for which the integration domains are one dimensional. We prove that any Gevrey series solution along the singular support of the system is the asymptotic expansion of a holomorphic solution given by a carefully chosen integral representation.Ministerio de Ciencia e InnovaciónFondo Europeo de Desarrollo RegionalJunta de Andalucí

Explicit calculations in rings of differential operators

We use the notion of a standard basis to study algebras of linear differential operators and finite type modules over these algebras. We consider the polynomial and the holomorphic cases as well as the formal case. Our aim is to demonstrate how to calculate classical invariants of germs of coherent (left) modules over the sheaf D of linear differential operators over Cn. The main invariants we deal with are: the characteristic variety, its dimension and the multiplicity of this variety at a point of the cotangent space. In the final chapter we shall study more refined invariants of D-modules linked to the question of irregularity: The slopes of a D-module along a smooth hypersurface of the base space.Dans ce cours on développe la notion de base standard, en vue d’étudier les algèbres d’opérateurs différentiels linéaires et les modules de type fini sur ces algèbres. On considère le cas des coefficients polynomiaux, des coefficients holomorphes ainsi que le cas des algèbres d’opérateurs à coefficients formels. Notre but est de montrer comment les bases standards permettent de calculer certains invariants classiques des germes de modules (à gauche) cohérents sur le faisceaux D des opérateurs différentiels linéaires sur Cn. Les principaux invariants que nous examinons sont : la variét´é caractéristique, sa dimension et sa multiplicité en un point du fibré cotangent. Dans le dernier chapitre nous étudions des invariants plus fins des D-modules qui sont reliés aux questions d’irrégularité : les pentes d’un D-module, le long d’une hypersurface lisse.Dirección General de Enseñanza Superior e Investigación CientíficaMinisterio de Ciencia y TecnologíaPlan Andaluz de Investigación (Junta de Andalucía

A flatness property for filtered D-modules

Let M be a coherent module over the ring DX of linear differential operators on an analytic manifold X and let Z1, · · · , Zk be k germs of transverse hypersurfaces at a point x ∈ X. The Malgrange-Kashiwara V-filtrations along these hypersurfaces, associated with a given presentation of the germ of M at x, give rise to a multifiltration U•(M) of Mx as in Sabbah’s paper [9] C. Sabbah, Proximité evanescente I. La structure polaire d’un D–module Compositio Math. 62 (1987) 283-319 and to an analytic standard fan in a way similar to [3] A. Assi., F. Castro-Jiménez and M. Granger, The analytic standard fan of a D-module, J. Pure Appl. Algebra 164 (2001) 3-21. We prove here that this standard fan is adapted to the multifiltration, in the sense of C. Sabbah. This result completes the proof of the existence of an adapted fan in [9] C. Sabbah, Proximité evanescente I. La structure polaire d’un D–module Compositio Math. 62 (1987) 283-319, for which the use of [8] C. Sabbah and F. Castro, Appendice à “proximité evanescente” I. La structure polaire d’un D–module, Compositio Math. 62 (1987) 320-328. is not possible

Gevrey expansions of hypergeometric integrals II

We study integral representations of the Gevrey series solutions of irregular hypergeometric systems under certain assumptions. We prove that, for such systems, any Gevrey series solution, along a coordinate hyperplane of its singular support, is the asymptotic expansion of a holomorphic solution given by a carefully chosen integral representation.Comment: 27 pages, 2 figure

Computing localizations iteratively

Let R = C[x] be a polynomial ring with complex coefficients and DX = Chx, ∂i be the Weyl algebra. Describing the localization Rf = R[f −1 ] for nonzero f ∈ R as a DX-module amounts to computing the annihilator A = Ann(f a ) ⊂ DX of the cyclic generator f a for a suitable negative integer a. We construct an iterative algorithm that uses truncated annihilators to build A for planar curves

Gevrey solutions of the irregular hypergeometric system associated with an affine monomial curve

We describe the Gevrey series solutions at singular points of the irregular hypergeometric system (GKZ system) associated with an affine monomial curve. We also describe the irregularity complex of such a system with respect to its singular support.Ministerio de Educación y CienciaJunta de Andalucí