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...

Seymour’s famous decomposition theorem for regular matroids states that any totally unimodular (TU) matrix can be constructed through a series of composition operations called k-sums starting from network matrices and their transposes and two compact representation matrices B1, B2 of a certain ten element matroid. Given that B1, B2 are binet matrices we examine the k-sums of network and binet m...

A square matrix is principally unimodular if every principal submatrix has determinant 0 or 1. Let A be a symmetric (0; 1)-matrix, with a zero diagonal. A PU-orientation of A is a skew-symmetric signing of A that is PU. If A 0 is a PU-orientation of A, then, by a certain decomposition of A, we can construct every PU-orientation of A from A 0. This construction is based on the fact that the PU-o...

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 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...

Loop transformations have been shown to be very useful in parallelising compilation and regular array design. This paper provides a solution to the open problem of automatic rewriting loop nests for non-unimodular loop transformations. We present an algorithm that rewrites a loop nest under any non-singular (unimodular or non-unimodular) transformation in a mechanical manner. The algorithm work...