On sparse representation of vectors in lattices and semigroups (Iskander Aliev, Cardiff University)

We will discuss the sparsity of the solutions to systems of linear Diophantine equations with and without non-negativity constraints. The sparsity of a solution vector is the number of its nonzero entries, which is referred to as the 0-norm of the vector. Our main results are new improved bounds on the minimal 0-norm of solutions to systems Ax=b, where A is an integer matrix, b is an integer vector and x is either a general integer vector (lattice case) or a non-negative integer vector (semigroup case). The talk is based on a joint work with G. Averkov, J. A. De Loera and T. Oertel.