Planar Configurations of Lattice Vectors and GKZ-Rational Toric Fourfolds in \Bbb P 6
Eduardo Cattani
and Alicia Dickenstein
DOI: 10.1023/B:JACO.0000022566.81227.03
Abstract
We introduce a notion of balanced configurations of vectors. This is motivated by the study of rational A-hypergeometric functions in the sense of Gelfand, Kapranov and Zelevinsky. We classify balanced configurations of seven plane vectors up to GL(2, 
)-equivalence and deduce that the only gkz-rational toric four-folds in 
6 are those varieties associated with an essential Cayley configuration. We show that in this case, all rational A-hypergeometric functions may be described in terms of toric residues. This follows from studying a suitable hyperplane arrangement.
)-equivalence and deduce that the only gkz-rational toric four-folds in 6 are those varieties associated with an essential Cayley configuration. We show that in this case, all rational A-hypergeometric functions may be described in terms of toric residues. This follows from studying a suitable hyperplane arrangement.Pages: 47–65
Keywords: $A$-hypergeometric functions; toric residues; balanced configurations; Cayley configurations
Full Text: PDF
References
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2. E. Cattani, D. Cox, and A. Dickenstein, “Residues in toric varieties,” Compositio Mathematica 108 (1997), 35-76.
3. E. Cattani, C. D'Andrea, and A. Dickenstein, “The A-hypergeometric system associated with a monomial curve,” Duke Math. J. 99 (1999), 179-207.
4. E. Cattani and A. Dickenstein, “A global view of residues in the torus,” Journal of Pure and Applied Algebra 117/118 (1997), 119-144.
5. E. Cattani, A. Dickenstein, and B. Sturmfels, “Rational hypergeometric functions,” Compositio Mathematica 128 (2001), 217-240.
6. E. Cattani, A. Dickenstein, and B. Sturmfels, “Binomial Residues,” Ann. Inst. Fourier 52 (2002), 687-708.
7. D. Cox, “Toric residues,” Arkiv f\ddot or Matematik 34 (1996), 73-96.
8. A. Dickenstein and B. Sturmfels, “Elimination theory in codimension two,” Journal of Symbolic Computation 34 (2002), 119-135.
9. A. Erdélyi et al., Higher Transcendental Functions, Based, in part, on notes left by Harry Bateman, McGraw- Hill Book Company, Inc., New York-Toronto-London, 1955.
10. I.M. Gel'fand, M. Kapranov, and A. Zelevinsky, “Generalized Euler integrals and A-hypergeometric functions,” Advances in Math. 84 (1990), 255-271.
11. I.M. Gel'fand, M. Kapranov, and A. Zelevinsky, Discriminants, Resultants and Multidimensional Determinants, Birkh\ddot auser, Boston, 1994.
12. I. M. Gel'fand, A. Zelevinsky, and M. Kapranov, “Hypergeometric functions and toral manifolds,” Functional Analysis and its Appl. 23 (1989), 94-106.
13. E. Hille, Ordinary Differential Equations in the Complex Domain, Dover, N.Y., 1997.
14. N. Ressayre, “Balanced configurations of 2n + 1 plane vectors,” Preprint (2002), arXiv:math.RA/0206234.
15. M. Saito, B. Sturmfels, and N. Takayama, “Hypergeometric polynomials and integer programming,” Compositio Mathematica 115 (1999), 185-204.
16. M. Saito, B. Sturmfels, and N. Takayama, Gr\ddot obner Deformations of Hypergeometric Differential Equations, Algorithms and Computation in Mathematics, Vol. 6, Springer-Verlag, Heidelberg, 1999.
17. A. Tsikh, Multidimensional Residues and Their Applications, American Math. Society, Providence, 1992.