ON NONNEGATIVE INTEGER MATRICES AND SHORT KILLING WORDS

Let n be a natural number, and letM be a set of nxn-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 is an element of M with M-1,M- ... M(n)5 = 0. This result has applications in automata theory and the theory of codes. Specifically, if X \subset \Sigma \ast is a finite incomplete code, then there exists a word w is an element of Sigma* of length polynomial in Sigma(x is an element of X)vertical bar x vertical bar such that w is not a factor of any word in X*. This proves a weak version of Restivo's conjecture.