Rational decomposition of polyhedra and a conjecture of Brlek and Reutenauer

We prove in the affirmative a conjecture of Brlek and Reutenauer (2003), which relates unambiguous rational sets to a well known result due to Khovanskii and Pukhlikov (1992) and independently to Lawrence (1990) on the valuations of the algebra spanned by the indicator functions of polyhedra.