Construction of regular languages and recognizability of polynomials

TL;DR

This paper explores how to construct numeration systems recognizing polynomial images of natural numbers using regular languages, focusing on the density function related to polynomial differences.

Contribution

It introduces a method to build numeration systems where polynomial images are recognizable by finite automata, extending the theory of regular languages and numeration systems.

Findings

Constructed numeration systems for polynomial images of N

Established conditions for regularity of polynomial image representations

Linked polynomial differences to density functions in regular languages

Abstract

A generalization of numeration system in which the set N of the natural numbers is recognizable by finite automata can be obtained by describing a lexicographically ordered infinite regular language. Here we show that if P belonging to Q[x] is a polynomial such that P(N) is a subset of N then we can construct a numeration system in which the set of representations of P(N) is regular. The main issue in this construction is to setup a regular language with a density function equals to P(n+1)-P(n) for n large enough.