# The range of non-linear natural polynomials cannot be context-free

## TL;DR

This paper proves that the set of natural values of a non-linear polynomial cannot have a context-free base-q representation, resolving a question about the limitations of context-free languages in describing polynomial-generated sets.

## Contribution

It establishes that only linear polynomials produce base-q representations that are context-free, introducing a new criterion combining the Interchange lemma and Pumping lemma for proving non-context-freeness.

## Key findings

- Non-linear polynomials do not produce context-free base-q representations.
- A new criterion for context-freeness is developed, combining existing lemmas.
- The result answers a previously open question by Shallit.

## Abstract

Suppose that some polynomial $f$ with rational coefficients takes only natural values at natural numbers, i.e., $L=\{f(n)\mid n\in \mathbb N\}\subset\mathbb N$. We show that the base-$q$ representation of $L$ is a context-free language if and only if $f$ is linear, answering a question of Shallit. The proof is based on a new criterion for context-freeness, which is a combination of the Interchange lemma and a generalization of the Pumping lemma.