Unimodular Covers of Multiples of Polytopes

Let P be a d-dimensional lattice polytope. We show that there exists a natural number cd , only depending on d, such that the multiples cP have a unimodular cover for every c ∈ N, c ≥ cd . Actually, an explicit upper bound for cd is provided, together with an analogous result for unimodular covers of rational cones. 1. STATEMENT OF RESULTS All polytopes and cones considered in this paper are assumed to be convex. A polytope P ⊂ Rd is called a lattice polytope, or integral polytope if its vertices belong to the standard lattice Zd . The multiplicity (or normalized volume) μ(P) of a d-dimensional lattice polytope (‘d-polytope’ for short) P ⊂ Rd is defined to be vol(P)/d!, where vol denotes the standard Euclidean volume; μ(P) is always an integral number. A simplex is called unimodular if it is a lattice simplex of multiplicity 1. A lattice simplex is called empty if its vertices are the only lattice points in it. Every unimodular simplex is empty, but the opposite implication is false in dimensions ≥ 3. (In dimension 2 the opposite implication is known as ‘Pick’s Theorem’.) For a (not necessarily integral) polytope P ⊂ Rd and a real number c ≥ 0 we let cP denote the image of P under the dilatation with factor c and center at the origin O ∈ Rd . The union of all unimodular simplices inside P will be denoted by Puni.cov. For a natural number d we let c d denote the infimum of the natural numbers c such that c′P = (c′P)uni.cov for all lattice d-polytopes P and all natural numbers c′ ≥ c. A priori, it is not excluded that c d = ∞ and, to the best of our knowledge, it has not been known up to now whether c d is finite except for the cases d = 1,2,3: c pol 1 = c pol 2 = 1 and c pol 3 = 2, where the first equation is trivial, the second is essentially Pick’s theorem, and a proof of the third can be found in Kantor and Sarkaria [KS]. The main result of this paper is the following upper bound, positively answering Problem 4 in [BGT2]: Theorem 1.1. For all natural numbers d > 1 one has