Mutsumi Saito

Grbner Deformations of Hypergeometric Differential Equations

The theory of Grbner bases is a main tool for dealing with rings of differential operators. This book reexamines the concept of Grbner bases from the point of view of geometric deformations. The algorithmic methods introduced in this book are particularly useful for studying the systems of multidimensional hypergeometric PDEs introduced by Gelfand, Kapranov, and Zelevinsky. A number of original research results are contained in the book, and many open problems are raised for future research in this rapidly growing area of computational mathematics.

Hypergeometric polynomials and integer programming

We examine connections between A-hypergeometric differential equations and the theory of integer programming. In the first part, we develop a ‘hypergeometric sensitivity analysis’ for small variations of constraint constants with creation operators and b-functions. In the second part, we study the indicial polynomial (b-function) along the hyperplane xi=0 via a correspondence between the optimal value of an integer programming problem and the roots of the indicial polynomial. Gröbner bases are used to prove theorems and give counter examples.

Irreducible quotients of A-hypergeometric systems

Gel’fand, Kapranov and Zelevinsky proved, using the theory of perverse sheaves, that in the Cohen–Macaulay case an A-hypergeometric system is irreducible if its parameter vector is non-resonant. In this paper we prove, using the theory of the ring of differential operators on an affine toric variety, that in general an A-hypergeometric system is irreducible if and only if its parameter vector is non-resonant. In the course of the proof, we determine the irreducible quotients of an A-hypergeometric system.

Noetherian properties of rings of differential operators of affine semigroup algebras

We consider the Noetherian properties of the ring of differential operators of an affine semigroup algebra. First we show that it is always right Noetherian. Next we give a condition, based on the data of the difference between the semigroup and its scored closure, for the ring of differential operators being anti-isomorphic to another ring of differential operators. Using this, we prove that the ring of differential operators is left Noetherian if the condition is satisfied. Moreover we give some other conditions for the ring of differential operators being left Noetherian. Finally we conjecture necessary and sufficient conditions for the ring of differential operators being left Noetherian.

The Spectrum of the Graded Ring of Differential Operators of a Scored Semigroup Algebra

We describe the set of ℤ d -graded prime ideals of the graded ring of the ring D of differential operators of a scored semigroup algebra. Moreover, we describe the characteristic varieties of ℤ d -graded critical D-modules of a certain type.

Critical Modules of the Ring of Differential Operators of an Affine Semigroup Algebra

Let D be the ring of differential operators of an affine semigroup algebra. Regarding the Krull dimension of finitely generated ℤ d -graded D-modules, we characterize critical ℤ d -graded D-modules. Moreover, we explicitly describe cyclic ones.

Rank versus Volume

Let A be a d × n-integer matrix of rank d and I A ⊂ K[∂] its toric ideal as before. Throughout this chapter we assume the homogeneity condition (3.3). This means that I A is a homogeneous ideal, i.e., I A is generated by binomials \(\partial _1^{{a_1}}\partial _2^{{a_2}}\cdot \cdot \cdot \partial _n^{{a_n}} - \partial _1^{{b_1}}\partial _2^{{b_2}}\cdot \cdot \cdot \partial _n^{{b_n}}\), where a 1+a 2+…+a n = b 1+b 2+…+b n . The convex hull of the columns of A is a polytope of dimension d − 1, denoted by conv(A), and its normalized volume is denoted by vol(A). Gel’fand, Kapranov and Zelevinsky found in their original work that the holonomic rank of the GKZ-hypergeometric system H A (β) is generally equal to the volume vol(A).

Integration of D-modules

Hypergeometric functions arise naturally from integrals of the form

Isomorphism Classes of A-Hypergeometric Systems

\Phi \left( x \right) = {\int_c {f\left( {x,t} \right)} ^\alpha }t_1^{{\gamma _1}}...t_m^{{\gamma _m}}d{t_1}...d{t_m},