The Cones Associated to Some Transversal Polymatroids

In this paper we describe the facets cone associated to transversal polymatroid presented by 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. 1 Preliminaries on polyhedral geometry An affine space generated by A ⊂ R is a translation of a linear subspace of R. If 0 = a ∈ R, then Ha will denote the hyperplane of R through the origin with normal vector a, that is, Ha = {x ∈ R | < x, a >= 0}, where <,> is the usual inner product in R. The two closed half spaces bounded by Ha are: H a = {x ∈ R | < x, a >≥ 0} and H− a = {x ∈ R | < x, a >≤ 0}. Recall that a polyhedral cone Q ⊂ R is the intersection of a finite number of closed subspaces of the form H a . If A = {γ1, . . . , γr} is a finite set of points in R the cone generated by A, denoted by R+A, is defined as R+A = { r ∑ i=1 aiγi | ai ∈ R+, with 1 ≤ i ≤ n}.