Irregular hypergeometric D-modules

We study the irregularity of hypergeometric D-modules $\mathcal{M}_A (\beta )$ via the explicit construction of Gevrey series solutions along coordinate subspaces in $X =\mathbb{C}^n$. As a consequence, we prove that along coordinate hyperplanes the combinatorial characterization of the slopes of $\mathcal{M}_A (\beta)$ given by M. Schulze and U. Walther in [21] still holds without any assumption on the matrix A. We also provide a lower bound for the dimensions of the spaces of Gevrey solutions along coordinate subspaces in terms of volumes of polytopes and prove the equality for very generic parameters. Holomorphic solutions outside the singular locus of $\mathcal{M}_A (\beta)$ can be understood as Gevrey solutions of order one along X at generic points and so they are included as a particular case.Comment: 41 pages; references, Remark 7.2. and 4 figures added; some comments changed; corrected typo

Characteristic cycles and Gevrey series solutions of AA-hypergeometric systems

We compute the $L$-characteristic cycle of an $A$-hypergeometric system and higher Euler-Koszul homology modules of the toric ring. We also prove upper semicontinuity results about the multiplicities in these cycles and apply our results to analyze the behavior of Gevrey solution spaces of the system.Comment: 22 page

Gevrey and formal Nilsson solutions of AA-hypergeometric systems

We prove that the space of Gevrey solutions of an $A$--hypergeometric system along a coordinate subspace is contained in a space of formal Nilsson solutions. Moreover, under some additional conditions, both spaces are equal. In the process we prove some other results about formal Nilsson solutions.Comment: 14 pages, 1 figure. Section 3 has been improved. In particular, new Theorem 3.3 and Theorem 3.4 replace Theorem 3.2 and Corollary 3.4 in the previous version, and we don't need now any condition on the parameters, except for the computation of the dimension of Gevrey series solutions of the hypergeometric syste

On the local monodromy of AA-hypergeometric functions and some monodromy invariant subspaces

We consider the p-center problem on tree graphs where the customers are modeled as continua subtrees. We address unweighted and weighted models as well as distances with and without addends. We prove that a relatively simple modification of Handler’s classical linear time algorithms for unweighted 1- and 2-center problems with respect to point customers, linearly solves the unweighted 1- and 2-center problems with addends of the above subtree customer model. We also develop polynomial time algorithms for the p-center problems based on solving covering problems and searching over special domains

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

Restriction of hypergeometric D-modules with respect to coordinate subspaces

We compute the restriction of an A-hypergeometric D-module with respect to a coordinate subspace under certain genericity conditions on the parameter.Ministerio de Educación y CienciaJunta de AndalucíaNational Science Foundatio

On the rank of an A-hypergeometric D-module versus the normalized volume of A

The rank of an $A$-hypergeometric $D$-module $M_A(\beta)$, associated with a full rank $(d\times n)$-matrix $A$ and a vector of parameters $\beta\in \mathbb{C}^d$, is known to be the normalized volume of $A$, denoted $\mathrm{vol}(A)$, when $\beta$ lies outside the exceptional arrangement $\mathcal{E}_(A)$, an affine subspace arrangement of codimension at least two. If $\beta\in \mathcal{E}(A)$ is simple, we prove that $d-1$ is a tight upper bound for the ratio $\mathrm{rank}(M_A(\beta))/\mathrm{vol}(A)$ for any $d\geq 3$. We also prove that the set of parameters $\beta$ such that this ratio is at least $2$ is an affine subspace arrangement of codimension at least $3$.Comment: 8 page

On irregular binomial D-modules

We prove that a holonomic binomial D–module MA(I, β) is regular if and only if certain associated primes of I determined by the parameter vector β ∈ Cd are homogeneous. We further describe the slopes of MA(I, β) along a coordinate subspace in terms of the known slopes of some related hypergeometric D–modules that also depend on β. When the parameter β is generic, we also compute the dimension of the generic stalk of the irregularity of MA(I, β) along a coordinate hyperplane and provide some remarks about the construction of its Gevrey solutions.Ministerio de Educación y CienciaMinisterio de Ciencia e InnovaciónFondo Europeo de Desarrollo RegionalJunta de AndalucíaUniversidad Complutense de Madri

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