Balanced Configurations of Lattice Vectors and GKZ-rational Toric Fourfolds in P⁶

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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,R) equivalence and deduce that the only gkz-rational toric four-folds in complex projective space P⁶ are those varieties associated with an essential Cayley configuration. In this case, we study a suitable hyperplane arrangement and show that all rational A-hypergeometric functions may be described in terms of toric residues.