Résumé
Let X be a smooth projective toric variety with k ample line bundles. Let Z be the zero locus of k generic sections. It is well known that the ambient quantum D-module of Z is cyclic i.e. is defined by an ideal of differential operators. In this paper, we give an explicit construction of this ideal as a quotient ideal of a GKZ system associated to the toric data of X and the line bundles. This description can be seen as a "left cancellation procedure". We consider some examples where this description enables us to compute generators of this ideal, and thus to give a presentation of the ambient quantum D-module.