Reflexive polytopes and the Picard ranks of Gorenstein toric Fano varieties

We prove that the sum of the Picard ranks of a polar pair of Gorenstein toric Fano varieties of dimension $d\ge 3$ is at most the minimum of the number of facets and vertices of the corresponding pair of reflexive polytopes minus $(d-1)$. This is a generalization of Eikelberg’s theory of affine dependences describing the Picard groups of toric varieties. The upper bound is achieved if and only if the polar pair is a simple-simplicial pair.