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?