Construction of regular languages and recognizability of polynomials

AbstractA generalization of numeration systems in which N is recognizable by finite automata can be obtained by describing a lexicographically ordered infinite regular language. We show that if P∈Q[x] is a polynomial such that P(N)⊂N then there exists a numeration system in which the set of representations of P(N) is regular. The main issue is to construct a regular language with a complexity function equals to P(n+1)−P(n) for n large enough