Sparse representation of vectors in lattices and semigroups

Abstract We study the sparsity of solutions to systems linear Diophantine equations with and without non-negativity constraints. The a solution vector is number its nonzero entries, which referred as $$\ell _0$$ ? 0 -norm vector. Our main results are new improved bounds on minimal $$A\varvec{x}=\varvec{b}$$ A x = b , where $$A\in \mathbb {Z}^{m\times n}$$ ? Z m × n $${\varvec{b}}\in {Z}^m$$ $$\varvec{x}$$ either general integer (lattice case) or non-negative (semigroup case). In certain cases, we give polynomial time algorithms for computing satisfying obtained bounds. show that our tight. can be seen functions naturally generalizing rank matrix over $$\mathbb {R}$$ R other subdomains such {Z}$$ . these rank-like all NP-hard compute in general, but polynomial-time computable fixed variables.