The Cones associated to some Transversal Polymatroids

In this paper we describe the facets cone associated to transversal polymatroid presented by $$\mathcal{A}$ = {{1,2},{2,3},…,{n-1,n},{n,1}}.$ Using the Danilov-Stanley theorem to characterize the canonicale module, we deduce that the base ring associated to this polymatroid is Gorenstein ring. Also, starting from this polymatroid we describe the transversal polymatroids with Gorenstein base ring in dimension 3 and with the help $Normaliz$ in dimension 4.