# Generalised polynomials and integer powers

## TL;DR

The paper proves that no generalized polynomial can precisely identify powers of two, and characterizes sequences that are both automatic and generalized polynomials, revealing limitations of such functions.

## Contribution

It establishes a non-existence result for generalized polynomials vanishing exactly on powers of two and characterizes sequences that are both automatic and generalized polynomials.

## Key findings

- No generalized polynomial vanishes only on powers of two.
- Sequences that are both automatic and generalized polynomials are fully characterized.
- Provides insights into the limitations of generalized polynomial functions.

## Abstract

We show that there does not exist a generalised polynomial which vanishes precisely on the set of powers of two. In fact, if $k \geq 2$ is and integer and $g \colon \mathbb{N} \to \mathbb{R}$ is a generalised polynomial such that $g(k^n) = 0$ for all $n \geq 0$ then there exists infinitely many $m \in \mathbb{N}$, not divisible by $k$, such that $g(mk^n) = 0$ for some $n \geq 0$. As a consequence, we obtain a complete characterisation of sequences which are simultaneously automatic and generalised polynomial.