We compute a stable polynomial matrix embedding a stabilizable one. The algorithm resembles the one described by Beelen and Van Dooren for the unimodular embedding problem. We also desribe the numerical problems associated with this kind of algorithms, and point out why the stable embedding algorithm has better prospects than its unimodular counterpart.

We prove an exponential upper bound for the number f(m,n) of all maximal triangulations of the m × n grid: f(m,n) < 2. In particular, this improves a result of S. Yu. Orevkov [1]. We consider lattice polygons P (with vertices in Z), for example the convex hull of the grid Pm,n := {0, 1, . . . , m} × {0, 1, . . . , n}. We want to estimate the number of maximal lattice triangulations of P , i.e.,...

Existence of a regular unimodular triangulation of the configuration + ∪ {(0, 0, . . . , 0)} in Rn, where + is the collection of the positive roots of a root system ⊂Rn and where (0, 0, . . . , 0) is the origin of Rn, will be shown for = Bn, Cn, Dn and BCn . Moreover, existence of a unimodular covering of a certain subconfiguration of the configuration An+1 will be studied.

We show that there is a unique extremal even unimodular lattice of dimension 48 which has an automorphism of order 5 of type 5− (8, 16)− 8. Since the three known extremal lattices do not admit such an automorphism, this provides a new example of an extremal even unimodular lattice in dimension 48.

Let (L,Q) be an even unimodular lattice, so L is a free Z-module of rank n, and Q : L → Z a positive definite regular integral quadratic form. Then L can be embedded into Euclidean n-space (R, (, )) with bilinear form defined by (x, y) := Q(x + y) − Q(x) − Q(y) and L defines a lattice sphere packing, whose density measures its error correcting properties. One of the main goals in lattice theory...