On Nonnegative Integer Matrices and Short Killing Words

Abstract

Let n be a natural number and M a set of n × n-matrices over the nonnegative integers such that the joint spectral radius of M is at most one. We show that if the zero matrix 0 is a product of matrices in M, then there are M 1 , . . . , M n 5 ∈ M with M 1 • • • M n 5 = 0. This result has applications in automata theory and the theory of codes. Specifically, if X ⊂ Σ * is a finite incomplete code, then there exists a word w ∈ Σ * of length polynomial in x∈X |x| such that w is not a factor of any word in X * . This proves a weak version of Restivo's conjecture.