Duke Mathematical Journal
- Duke Math. J.
- Volume 151, Number 3 (2010), 385-429.
We study quotients of the Weyl algebra by left ideals whose generators consist of an arbitrary -graded binomial ideal in along with Euler operators defined by the grading and a parameter . We determine the parameters for which these -modules (i) are holonomic (equivalently, regular holonomic, when is standard-graded), (ii) decompose as direct sums indexed by the primary components of , and (iii) have holonomic rank greater than the rank for generic . In each of these three cases, the parameters in question are precisely those outside of a certain explicitly described affine subspace arrangement in . In the special case of Horn hypergeometric -modules, when is a lattice-basis ideal, we furthermore compute the generic holonomic rank combinatorially and write down a basis of solutions in terms of associated -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 [DMM]
Duke Math. J., Volume 151, Number 3 (2010), 385-429.
First available in Project Euclid: 8 February 2010
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 33C70: Other hypergeometric functions and integrals in several variables 32C38: Sheaves of differential operators and their modules, D-modules [See also 14F10, 16S32, 35A27, 58J15]
Secondary: 14M25: Toric varieties, Newton polyhedra [See also 52B20] 13N10: Rings of differential operators and their modules [See also 16S32, 32C38]
Dickenstein, Alicia; Matusevich, Laura Felicia; Miller, Ezra. Binomial $D$ -modules. Duke Math. J. 151 (2010), no. 3, 385--429. doi:10.1215/00127094-2010-002. https://projecteuclid.org/euclid.dmj/1265637658