Let n be a natural number and M a set of nΓn-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 M1β,β¦,Mn5ββM with M1ββ―Mn5β=0. This result has applications in
automata theory and the theory of codes. Specifically, if XβΞ£β
is a finite incomplete code, then there exists a word wβΞ£β of
length polynomial in βxβXββ£xβ£ such that w is not a factor of any
word in Xβ. This proves a weak version of Restivo's conjecture.Comment: This version is a journal submission based on a STACS'19 paper. It
extends the conference version as follows. (1) The main result has been
generalized to apply to monoids generated by finite sets whose joint spectral
radius is at most 1. (2) The use of Carpi's theorem is avoided to make the
paper more self-contained. (3) A more precise result is offered on Restivo's
conjecture for finite code