The flag polynomial of the Minkowski sum of simplices

TL;DR

This paper introduces the flag polynomial for polytopes, especially for Minkowski sums of simplices, providing a canonical expression and demonstrating its utility through formulas and face chain comparisons.

Contribution

It defines the flag polynomial for Minkowski sums of simplices and expresses it via the master polytope, enabling direct computations and new insights.

Findings

Derived a formula for the f-polynomial of Minkowski sums of two simplices

Calculated the maximum number of d-faces in such sums

Quantified the discrepancy in face chains between Minkowski sums and simple polytopes

Abstract

For a polytope we define the {\em flag polynomial}, a polynomial in commuting variables related to the well-known flag vector and describe how to express the the flag polynomial of the Minkowski sum of $k$ standard simplices in a direct and canonical way in terms of the {\em $k$-th master polytope} $P(k)$ where $k\in\nats$. 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.