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 \(f:S\rightarrow \mathbb {C}\), where \(S=\mathcal {A}^*\) is the monoid of words based on an arbitrary alphabet \(\mathcal {A}\) under concatenation, and we prove that if there exists a monoid S such that \(\mathcal{S}\mathcal{P}(S,\mathbb {C})\ne \mathcal {P}(S,\mathbb {C})\), then also \(\mathcal{S}\mathcal{P}(\mathcal {A}^*,\mathbb {C})\ne \mathcal {P}(\mathcal {A}^*,\mathbb {C})\) for a certain alphabet \(\mathcal {A}\). We propose as an open problem to prove or disprove that \(\mathcal{S}\mathcal{P}(\mathcal {A}^*,\mathbb {C})=\mathcal {P}(\mathcal {A}^*,\mathbb {C})\) for arbitrary alphabets \(\mathcal {A}\). Finally, we apply our algebraic framework to combinatorial objects by proving that permutation pattern functions are polynomial maps. Our results are motivated by previous work of Shulman [3, 4].