Logarithm-free \(A\)-hypergeometric series. (English) Zbl 1031.33011

Saito, Mutsumi

Duke Math. J. 115, No. 1, 53-73 (2002).

The author studies the solutions of an \(A\)-hypergeometric system cf. I. M. Gel’fand, A. V. Zelevinskij and M. M. Kapranov [Sov. Math. Dokl. 37, No. 3, 678-682 (1988; Zbl 0667.33010)]. He obtains an explicit formula for the vector space of logarithm-free \(A\)-hypergeometric series in functions of the volumes of some facets. It is proved that there exists a fundamental system of solutions in the case where \(\text{conv}(A)\) is a simplex and, in this case, there is given a rank formula for the system. The author also obtains a classification of \(A\)-hypergeometric systems as analytic \({\mathcal D}\)-modules.

Reviewer: Doru Ştefănescu (Bucureşti)

MSC:

33C70	Other hypergeometric functions and integrals in several variables
14M25	Toric varieties, Newton polyhedra, Okounkov bodies
13N10	Commutative rings of differential operators and their modules
16S32	Rings of differential operators (associative algebraic aspects)

Keywords:

\(A\)-hypergeometric series

Citations:

Zbl 0667.33010

References:

[1]	A. Adolphson, Hypergeometric functions and rings generated by monomials , Duke Math. J. 73 (1994), 269–290. · Zbl 0804.33013
[2]	E. Cattani, C. D’Andrea, and A. Dickenstein, The \(\mathcal A\)-hypergeometric system associated with a monomial curve , Duke Math. J. 99 (1999), 179–207. · Zbl 0952.33009
[3]	I. M. Gel’fand, A. V. Zelevinskiĭ, and M. M. Kapranov, Equations of hypergeometric type and Newton polyhedra , Soviet Math. Dokl. 37 , no. 3 (1988), 678–682. · Zbl 0667.33010
[4]	–. –. –. –., Hypergeometric functions and toric varieties , Funct. Anal. Appl. 23 (1989), 94–106. · Zbl 0721.33006
[5]	L. F. Matusevich, Rank jumps in codimension 2 \(A\)-hypergeometric systems , J. Symbolic Comput. 32 (2001), 619–641. \CMP1 866 707 · Zbl 1074.32004
[6]	M. Saito, Isomorphism classes of \(A\)-hypergeometric systems , Compositio Math. 128 (2001), 323–338. \CMP1 858 340 · Zbl 1075.33009
[7]	M. Saito, B. Sturmfels, and N. Takayama, Gröbner Deformations of Hypergeometric Differential Equations , Algorithms Comput. Math. 6 , Springer, Berlin, 2000. · Zbl 0946.13021
[8]	M. Saito and W. N. Traves, “Differential algebras on semigroup algebras” in Symbolic Computation: Solving Equations in Algebra, Geometry, and Engineering (South Hadley, Mass., 2000) , Contemp. Math. 286 , Amer. Math. Soc., Providence, 2001, 207–226. \CMP1 874 281 · Zbl 1058.16026
[9]	A. Schrijver, Theory of Linear and Integer Programming , Wiley-Intersci. Ser. Discrete Math., Wiley, Chichester, England, 1986. · Zbl 0665.90063