Is a smooth function sending algebraic numbers to algebraic numbers a polynomial?

In other words if $f:\mathbb{C} \to \mathbb{C}$ is an entire function such that $$f(\overline{\mathbb{Q}}) \subset \overline{\mathbb{Q}}$$ where $\overline{\mathbb{Q}}$ is the field of algebraic numbers. Can we say that it is a polynomial?

A similar question (though I'm not sure if it is the same question): Let $g$ be a meromorphic function such that $g:\mathbb{\overline{\mathbb{Q}}} \to \overline{\mathbb{Q}}\cup \{\infty\}$; can we say it is a rational function?

This is a CW answer to remove this question from the unanswered list -- this question has been answered on Mathoverflow. The answer is no; see this paper.