Polynomial maps on the monoid of words

Abstract

We briefly visit the theory of polynomial and semipolynomial maps defined on an arbitrary monoid, with range a commutative group. Then we characterize 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 an arbitrary alphabet $\mathcal{A}$ under concatenation, and we use this characterization to prove that if there exists a monoid $S\not\in\mathcal{CS}$ such that $\mathcal{SP}(S,\mathbb{C})\neq \mathcal{P}(S,\mathbb{C})$, then also $\mathcal{SP}(\mathcal{A}^*,\mathbb{C})\neq\mathcal{P}(\mathcal{A}^*,\mathbb{C})$ for a certain alphabet $\mathcal{A}$. We propose as an open problem to prove or disprove that $\mathcal{SP}(\mathcal{A}^*,\mathbb{C})=\mathcal{P}(\mathcal{A}^*,\mathbb{C})$ for arbitrary alphabets $\mathcal{A}$. Our results are motivated by previous work of Shulman.