Growth functions associated with Artin monoids of finite type

Abstract

We prove that the growth functions associated with Artin-monoids of finite type are rational functions whose numerators is equal to 1. We give an explicit formula for the denominator polynomial $N_{M}(t)$ and give three conjectures on it: 1. $N_{M}(t)$ is irreducible up to a factor 1-t, 2. there are $l$-1 real distinct roots of $N_{M}(t)$ on the interval (0,1), and 3. the smallest real root on (0,1) is the unique smallest absolute values of all roots of $N_{M}(t)$.