Dimension of spaces of polynomials on abelian topological semigroups

In this paper we study (continuous) polynomials $p: J\to X$, where $J$ is an abelian topological semigroup and $X$ is a topological vector space. If $J$ is a subsemigroup with non-empty interior of a locally compact abelian group $G$ and $G=J-J$, then every polynomial $p$ on $J$ extends uniquely to a polynomial on $G$. It is of particular interest to know when the spaces $P^n (J,X)$ of polynomials of order at most $n$ are finite dimensional. For example we show that for some semigroups the subspace $P^n_{R} (J,\mathbf{C})$ of Riss polynomials (those generated by a finite number of homomorphisms $\alpha: J\to \mathbf{R}$) is properly contained in $P^n (G,\mathbf{C})$. However, if $P^1 (J,\mathbf{C})$ is finite dimensional then $P^n_{R} (J,\mathbf{C})= P^n (J,\mathbf{C})$. Finally we exhibit a large family of groups for which $P^n (G,\mathbf{C})$ is finite dimensional.