2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/97703We give equivalent and sufficient criteria for the automorphism group of a complete toric variety, respectively a Gorenstein toric Fano variety, to be reductive. In particular we show that the automorphism group of a Gorenstein toric Fano variety is reductive, if the barycenter of the associated reflexive polytope is zero. Furthermore a sharp bound on the dimension of the reductive automorphism group of a complete toric variety is proven by studying the set of Demazure roots.AMS-LaTeX, 20 pages with 1 figureAlgebraic GeometryCombinatorics14J45; 14J50; 14M25; 52B20text