The Flag Polynomial of the Minkowski Sum of Simplices

For a polytope we define the flag polynomial, a polynomial in commuting variables related to the well-known flag vector and describe how to express the flag polynomial of the Minkowski sum of k standard simplices in a direct and canonical way in terms of the k-th master polytope P(k) where ${k \in \mathbb {N}}$ . The flag polynomial facilitates many direct computations. To demonstrate this we provide two examples; we first derive a formula for the f -polynomial and the maximum number of d-dimensional faces of the Minkowski sum of two simplices. We then compute the maximum discrepancy between the number of (0, d)-chains of faces of a Minkowski sum of two simplices and the number of such chains of faces of a simple polytope of the same dimension and on the same number of vertices.