We derive a mass formula for n-dimensional unimodular lattices having any prescribed root system. We use Katsurada’s formula for the Fourier coefficients of Siegel Eisenstein series to compute these masses for all root systems of even unimodular 32-dimensional lattices and odd unimodular lattices of dimension n ≤ 30. In particular, we find the mass of even unimodular 32dimensional lattices with...

The coordination sequence of a lattice L encodes the word-length function with respect toM , a set that generates L as a monoid. We investigate the coordination sequence of the cyclotomic lattice L = Z[ζm], where ζm is a primitive m th root of unity and where M is the set of all m roots of unity. We prove several conjectures by Parker regarding the structure of the rational generating function ...

The extent to which a sequence of finite length differs from a shifted version of itself is measured by its aperiodic autocorrelations. Of particular interest are sequences whose entries are 1 or −1, called binary sequences, and sequences whose entries are complex numbers of unit magnitude, called unimodular sequences. Since the 1950s, there is sustained interest in sequences with small aperiod...

A (0, 1) matrix A is strongly unimodular if A is totally unimodular and every matrix obtained from A by setting a nonzero entry to 0 is also totally unimodular. Here we consider the linear discrepancy of strongly unimodular matrices. It was proved by Lováz, et.al. [5] that for any matrix A, lindisc(A) ≤ herdisc(A). (1) When A is the incidence matrix of a set-system, a stronger inequality holds:...

We show that the Ehrhart h-vector of an integer Gorenstein polytope with a unimodular triangulation satisfies McMullen’s g-theorem; in particular it is unimodal. This result generalizes a recent theorem of Athanasiadis (conjectured by Stanley) for compressed polytopes. It is derived from a more general theorem on Gorenstein affine normal monoids M: one can factor K[M] (K a field) by a “long” re...

We revisit the construction method of even unimodular lattices using ternary self-dual codes given by the third author (M. Ozeki, in Théorie des nombres, J.-M. De Koninck and C. Levesque (Eds.) (Quebec, PQ, 1987), de Gruyter, Berlin, 1989, pp. 772–784), in order to apply the method to odd unimodular lattices and give some extremal (even and odd) unimodular lattices explicitly. In passing we cor...

We give an incremental polynomial time algorithm for enumerating the vertices of any polyhedron P(A, 1 ̄ ) = {x ∈ R | Ax ≥ 1 ̄ , x ≥ 0 ̄ }, when A is a totally unimodular matrix. Our algorithm is based on decomposing the hypergraph transversal problem for unimodular hypergraphs using Seymour’s decomposition of totally unimodular matrices, and may be of independent interest.

A polynomial f ∈ C[z] is unimodular if all its coefficients have unit modulus. Let Un denote the set of unimodular polynomials of degree n−1, and let Un denote the subset of reciprocal unimodular polynomials, which have the property that f(z) = ωzn−1f(1/z) for some complex number ω with |ω| = 1. We study the geometric and arithmetic mean values of both the normalized Mahler’s measure M(f)/ √ n ...

Unimodular Substitutions on 2 letters. Conjecture: the dynamical system associated with a primitive substitution on 2 letters, with matrix in SL(2,Z), is measurably isomorphic to a circle rotation. There is a very convenient criterium, due to B.Host: Definition: the substitution σ has strong coincidence if there exists n and k such that σ(0) and σ(1) have same letter of index k, and the 2 prefi...