arXiv subjects
Publications and source records attributed to F. J. Castro-Jimenez.
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.
We consider the Weyl algebra A (=A_n(k)) and its Rees algebra B with respect to the Bernstein filtration. The homogenisation of a differential operator in A is an element in B. In this paper we establish the validity of the division theorem for homogenized differential operators and Buchberger's algorithm for computing Groebner (or standard) bases in B. As an application we describe an algorithm for computing delta-standard bases in the Weyl algebra A.
We describe the Gevrey series solutions at singular points of the irregular hypergeometric system (GKZ system) associated with an affine plane monomial curve. We also describe the irregularity complex of such a system with respect to its singular support.
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. We use restriction and some results in D-module theory to reduce our study to the two dimensional case.
Let $O_X$ (resp. $D_X$) be the sheaf of holomorphic functions (resp. the sheaf of linear differential operators with holomorphic coefficients) on $X$ (=the complex affine n-space). Let $Y$ be a locally weakly quasi-homogeneous free divisor defined by a polynomial $f$. In this paper we prove that, locally, the annihilating ideal of $1/f^k$ over $D_X$ is generated by linear differential operators of order 1 (for $k$ big enough). For this purpose we prove a vanishing theorem for the extension groups of a certain logarithmic $D_X$--module with $O_X$. The logarithmic $D_X$--module is naturally associated with $Y$. This result is related to the so called Logarithmic Comparison Theorem.
Let M be a coherent module over the ring D of linear differential operators on an analytic manifold X and let us consider k germs of transverse hypersurfaces at a point x in X. The Malgrange-Kashiwara V-filtrations along these hypersurfaces, associated with a given presentation of the germ of M at the point x, give rise to a multifiltration U(M) of M_x introduced by Sabbah and to an analytic standard fan as developed by Assi-Castro-Granger. 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], for which the use of [8] is not possible (see References).
Let $D$ be a divisor in ${\bf C}^n$. We present methods to compare the ${\mathcal D}$-module of the meromorphic functions ${\mathcal O}[* D]$ to some natural approximations. We show how the analytic case can be treated with computations in the Weyl algebra.
We compute, by D-module restrictions, the slopes of irregular hypergeometric systems associated to a monomial curve.
The relationship between ${\cal D}$-modules and free divisors has been studied in a general setting by L. Narváez and F.J. Calderón. Using the ideas of these works we prove in this article a duality formula between two ${\cal D}$-modules associated to a class of free divisors on ${\bf C}^n$ and we give some applications