I'm now trying to develop some protocols to work with cryptography over the ring $\mathbb{Z}_{2^k}$, and I tried to find a ring version of the Schwartz Zippel lema. The main idea is to work in a ring extension or Galois ring, since this decreases greatly the ratio of roots/ring size, nonetheless, I was wondering if by working with randomised polynomials I could get a similar result, or combine both ideas together to reduce the degree of the extension and end up with the same security parameter.
The main question is, what is the distribution on the amount of roots of a randomised polynomial over the ring $\mathbb{Z}_{2^k}$? Is there somewhere I could find the expected value for this result?