Is $Th(\mathbb{Z}[x])$ uncountably categorical? 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. 

Since you contain the arithmetic of the integers you can pick some infinite family of primes and say that an element of degree $0$ is divisible by those primes and not any other primes. This gives uncountably many types over a countable set, so  the theory is not $\aleph_0$ stable and thus not uncountably categorical.
