Se p 19 97 On the width of lattice - free simplices

Integral polytopes (see [Z ]for the basic definitions) are of interest in combinatorics, linear programming, algebraic geometry-toric varieties [D,O], number theory [K-L.]. We study here lattice-free simplices, that is simplices intersecting the lattice only at their vertices. A natural question is to measure the " flatness " of these polytopes, with respect to integral dual vectors. This (arithmetical) notion plays a crucial role :-in the classification (up to affine unimodular maps) of lattice-free simplices in dimension 3 (see [O,MMM].-in the construction of a polynomial-time algorithm for integral linear programming (flatness permits induction on the dimension, [K-L]). Unfortunately there were no known examples (in any dimension) of lattice-free poly-topes with width bigger than 2 .We prove here the following Theorem: Given any positive number α strictly inferior to 1 e , for d large enough there exists a lattice-free simplex of dimension d and width superior to αd. The proof is non-constructive and uses replacing the search for lattice-free simplices in Z d by the search for " lattice-free lattices " containg Z d (" turning the problem inside out " ,see par. II),specializing in the next step to lattices of a simple kind,depending on a prime number p.The existence of lattice-free simplices with big width is then deduced by elementary computations,through a sufficient inequality involving the dimension d,the width k and the prime p (see (14)). The author thanks with pleasure H. Lenstra for crucial suggestions, V. Guillemin and I. Bernstein for comments. Notations P d : The set of integral polytopes in R d ; if P is such a polytope,P is a convex compact set,the set Vert(P) of vertices of P is a subset of Z d. S d : The set of integral simplices in R d. In particular σ d will denote the canonical simplex with vertices at the origin and