Duke Mathematical Journal

Binomial D-modules

Alicia Dickenstein, Laura Felicia Matusevich, and Ezra Miller

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We study quotients of the Weyl algebra by left ideals whose generators consist of an arbitrary Zd-graded binomial ideal I in C[1,,n] along with Euler operators defined by the grading and a parameter βCd. We determine the parameters β 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 rank for generic β. In each of these three cases, the parameters in question are precisely those outside of a certain explicitly described affine subspace arrangement in Cd. 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 [DMM]

Article information

Duke Math. J., Volume 151, Number 3 (2010), 385-429.

First available in Project Euclid: 8 February 2010

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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

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