On polytopes associated to factorisations of prime-powers

1: We study polytopes associated to factorisations of prime powers. These polytopes have explicit descriptions either in terms of their vertices or as intersections of closed halfspaces associated to their facets. We give formulae for their f−vectors. 1 Main results Polytopes have two dual descriptions: They can be given either as convex hulls of finite sets or as compact sets of the form ∩f∈Ff (R+) where F is a finite set of affine functions and where f(R+) denotes the closed half-space on which the affine function f is non-negative. It is difficult to construct families of polytopes where both descriptions are explicit. The aim of this paper is to study a new family of such examples. These polytopes are associated to vector-factorisations of primepowers where a d−dimensional vector-factorisation of a prime-power pe is an integral vector (v1, v2, . . . , vd) ∈ N d such that pe = v1 · v2 · · · vd. Given a prime power pe ∈ N and a natural integer d ≥ 1, we denote by P(pe, d) the convex hull of all d−dimensional vector-factorisations (v1, v2, . . . , vd) ∈ N d of pe. The case e = 0 yields the unique vector-factorisation (1, 1, . . . , 1) and is without interest. For d = 2 and e ≥ 2 the polytope P(pe, 2) is a 2−dimensional polygon with vertices (1, pe), (p, pe−1), . . . , (pe−1, p), (1, pe). For e = 1, the polytope P(p, d) is a (d − 1)−dimensional simplex with vertices (p, 1, . . . , 1), (1, p, 1, . . . , 1), . . . , (1, . . . , 1, p). The observation that the combinatorial properties of P(pe, d) are independent of the prime p in these examples is a general fact: The combinatorial properties of the polytope P(pe, d) are independent always independent of the prime number p. It is in fact possible to replace every occurence of p by an arbitrary real constant which is strictly greater than 1. (The choice of a strictly positive real number which is strictly smaller than 1 leads to a