Smith Normal Form over Local Rings and Related Problems

Abstract

The Smith normal form is a diagonalization of matrices with many applications in diophantine analysis, graph theory, system control theory, simplicial homology, and more recently, in topological analysis of big data. Efficient computation of Smith normal form is a well-studied area for matrices with integer and polynomial entries. Existing successful algorithms typically rely on elimination for dense matrices and iterative Krylov space methods for sparse matrices.

Our interest lies in computing Smith normal form for sparse matrices over local rings, where traditional iterative methods face challenges due to the lack of unique minimal polynomials. We explore different approaches to tackling this problem for two local rings: the integers modulo a prime power, and the polynomials modulo a power of an irreducible polynomial. Over local polynomial rings, we find success in linearization into larger dimension matrices over the base field. Effectively we transform the problem of computing the Smith normal form into a small number of rank problems over the base field. The latter problem has existing efficient algorithms for sparse and dense matrices.

The problem is harder over local integer rings. We take the approach of hybrid sparse-dense algorithms. We also tackle a restricted version of the problem where we detect only the first non-trivial invariant factor. We also give an algorithm to find the first few invariant factors using iterative rank-1 updates. This method becomes dense when applied to finding all the invariant factors.

We digress slightly into the related problem of preconditioning. We show that linear- time preconditioners are suitable for computing Smith normal form, and computing nullspace samples. For the latter problem we design an algorithm for computing uniform samples from the nullspace.

On a separate track, we focus on the properties of the Smith normal form decomposition. We relate the invariant factors to the eigenvalues. Our ultimate goal is to extend the applications of numerical algorithms for computing eigenvalues to computing the invariant factors of symbolic matrices.