On polynomial functions on semigroups

Given a semigroup $\mathcal{S}$ and a divisible Abelian group $\mathcal{H}$ we call a function $f: \mathcal{S} \longrightarrow \mathcal{H}$ a polynomial if $ \Delta_{h_m} \cdots \Delta_{h_1}f=0 $ for some $ m \in \mathbb{N} $ and any $ h_1,\ldots, h_m \in \mathcal{S},$ and we refer to $f$ as a semi-polynomial if $\Delta_h^n f=0$ for some $ n \in \mathbb{N}$ and every $h \in \mathcal{S}.$ A long list of papers, starting from a work of Fréchet (1909) up to the present time, have been devoted to the comparison of these two classes of mappings. In the talk we will discuss the present state of the topic and show that the classes of polynomials and semi-polynomials coincide for a wide variety of semigroups including all groups and all commutative semigroups.