Let $X=\{f\in \mathbb{C}[z]\mid |z| \neq 1 \implies f(z) \neq 0\} $.

The motivation for consideration of such an $X$ is the the concept of Lee-Yang polynomials.

With the standard multiplication, $X$ is an Abelian semigroup with cancellation property.

Let $G$ be the Grothendieck group associated with $X$.

Is there a well known group which is isomorphic to $G$? In other words, is there an alternative formulation of $G$ in terms of some well known group? Is there a natural topology on $G$ which makes it a locally compact topological group?