Some algebraic aspects of mesoprimary decomposition

Recent results of Kahle and Miller give a method of constructing primary decompositions of binomial ideals by first constructing "mesoprimary decompositions" determined by their underlying monoid congruences. Monoid congruences (and therefore, binomial ideals) can present many subtle behaviors that must be carefully accounted for in order to produce general results, and this makes the theory complicated. In this paper, we examine their results in the presence of a positive $A$-grading, where certain pathologies are avoided and the theory becomes more accessible. Our approach is algebraic: while key notions for mesoprimary decomposition are developed first from a combinatorial point of view, here we state definitions and results in algebraic terms, which are moreover significantly simplified due to our (slightly) restricted setting. In the case of toral components (which are well-behaved with respect to the $A$-grading), we are able to obtain further simplifications under additional assumptions. We also provide counterexamples to two open questions, identifying (i) a binomial ideal whose hull is not binomial, answering a question of Eisenbud and Sturmfels, and (ii) a binomial ideal $I$ for which $I_\text{toral}$ is not binomial, answering a question of Dickenstein, Miller and the first author

Binomial D-modules

We study quotients of the Weyl algebra by left ideals whose generators consist of an arbitrary Z^d-graded binomial ideal I along with Euler operators defined by the grading and a parameter in C^d. We determine the parameters for which these D-modules (i) are holonomic (equivalently, regular holonomic, when I is standard-graded); (ii) decompose as direct sums indexed by the primary components of I; and (iii) have holonomic rank greater than the generic rank. In each of these three cases, the parameters in question are precisely those outside of a certain explicitly described affine subspace arrangement in C^d. In the special case of Horn hypergeometric D-modules, when I is a lattice basis ideal, we furthermore compute the generic holonomic rank combinatorially and write down a basis of solutions in terms of associated A-hypergeometric functions. This study relies fundamentally on the explicit lattice point description of the primary components of an arbitrary binomial ideal in characteristic zero, which we derive in our companion article arxiv:0803.3846.Comment: This version is shorter than v2. The material on binomial primary decomposition has been split off and now appears in its own paper arxiv:0803.384

Nilsson solutions for irregular A-hypergeometric systems

We study the solutions of irregular A-hypergeometric systems that are constructed from Gr\"obner degenerations with respect to generic positive weight vectors. These are formal logarithmic Puiseux series that belong to explicitly described Nilsson rings, and are therefore called (formal) Nilsson series. When the weight vector is a perturbation of (1,...,1), these series converge and provide a basis for the (multivalued) holomorphic hypergeometric functions in a specific open subset of complex n-space. Our results are more explicit when the parameters are generic or when the solutions studied are logarithm-free. We also give an alternative proof of a result of Schulze and Walther that inhomogeneous A-hypergeometric systems have irregular singularities.Comment: Terminology changed: see Definition 2.6 in current version. Corrections made to Theorem 6.6, Corollary 6.7 and Corollary 6.8 in version 1 (now Theorem 6.7, Corollary 6.9 and Corollary 6.10, respectively). Added Corollary 6.3 and Example 6.8. Some stylistic changes, some typos correcte