Abstract
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).
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© 2000 Springer-Verlag Berlin Heidelberg
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Saito, M., Sturmfels, B., Takayama, N. (2000). Rank versus Volume. In: Gröbner Deformations of Hypergeometric Differential Equations. Algorithms and Computation in Mathematics, vol 6. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-04112-3_4
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DOI: https://doi.org/10.1007/978-3-662-04112-3_4
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-08534-5
Online ISBN: 978-3-662-04112-3
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