2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/100926Let X be a complex, Gorenstein, Q-factorial, toric Fano variety. We prove two conjectures on the maximal Picard number of X in terms of its dimension and its pseudo-index, and characterize the boundary cases. Equivalently, we determine the maximal number of vertices of a simplicial reflexive polytope.Final version, to appear in Annales de l'Institut FourierAlgebraic GeometryCombinatorics52B20, 14M25, 14J45text