Polyhedral Omega is a new algorithm for solving linear Diophantine systems (LDS), i.e., for computing a multivariate rational function representation of the set of all non-negative integer solutions to a system of linear equations and inequalities. Polyhedral Omega combines methods from partition analysis with methods from polyhedral geometry. In particular, we combine MacMahon’s iterative appr...

The goal of this paper is to generalize the main results of [KM1] and subsequent papers on metric Diophantine approximation with dependent quantities to the set-up of systems of linear forms. In particular, we establish ‘joint strong extremality’ of arbitrary finite collection of smooth nondegenerate submanifolds of R. The proofs are based on quantitative nondivergence estimates for quasi-polyn...

LinBox[6] is a C++ template library for high-performance exact linear algebra. It provides optimized facilities for solving rational systems, and computing invariants and canonical forms of linear operators. It acts as middleware on top of existing low-level software libraries for multiprecision integer arithmetic (GMP, NTL), finite field algebra (Givaro, NTL) and linear algebra (BLAS, ATLAS). ...

In this paper we introduce the version 2.5 of Normaliz , a program for the computation of Hilbert bases of rational cones and the normalizations of affine monoids. It may also be used for solving diophantine linear systems of inequalities, equations and congruences. We present some of the new features of the program, as well as some recent achievements.

– In Diophantine Approximation, inhomogeneous problems are linked with homogeneous ones by means of the so-called Transference Theorems. We revisit this classical topic by introducing new exponents of Diophantine approximation. We prove that the exponent of approximation to a generic point in R n by a system of n linear forms is equal to the inverse of the uniform homogeneous exponent associate...

In Diophantine Approximation, inhomogeneous problems are linked with homogeneous ones by means of the so-called Transference Theorems. We revisit this classical topic by introducing new exponents of Diophantine approximation. We prove that the inhomogeneous exponent of approximation to a generic point in R by a system of n linear forms is equal to the inverse of the uniform homogeneous exponent...

Let R be a ring and let (a1,…,an)∈Rn unimodular vector, where n≥2 each ai is in the center of R. Consider linear equation a1X1+⋯+anXn=0, with solution set S. Then S=S1+⋯+Sn, Si naturally derived from (a1,…,an), we give presentation S terms generators taken appropriate relations. Moreover, under suitable assumptions, elucidate structure quotient module S/Si. Furthermore, assuming that principal ...