Consider $T=Th(\mathbb{Z}[x])$ in the language $L = \{0,1,+,\times,deg(), \circ\}$ where $0,1,+$ and $\times$ have their usual interpretations, $deg()$ is a unary function symbol which gives the degree of a polynomial and $\circ$ is a binary function symbols where (if $p(x)$ and $q(x)$ are polynomials) $$ p(x)\circ q(x) = p(q(x))$$ and if $p(x)$ is a "constant", then $$p(x) \circ q(x) = p(x)$$
Clearly, $T$ is not countably categorical since it has $\aleph_0$ many $1-$types definable without parameters. However, I cannot figure out whether the theory is uncountably categorical.
Thank you.
