  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.
