Polynomial maps on the monoid of words

We study the space $\mathcal{P}(S,\mathbb{C})$ of polynomial maps $f:S\to \mathbb{C}$, where $S=\mathcal{A}^*$ is the monoid of words based on a finite alphabet $\mathcal{A}$ under concatenation. To motivate the study of this space, we first briefly visit the theory of polynomial and semipolynomial maps defined on an arbitrary monoid, with range a commutative group. Our results are motivated by previous work of Shulman.