Let $R$ be an integral domain with maximal ideal $M$, and suppose $R/M$ is algebraically closed. Let $\phi: R[x_0, ..., x_n] \to (R/M)[x_0, ..., x_n]$ defined by taking the coefficients modulo $M$. What does the preimage of the maximal ideal $I = (x_0 - \beta_0 + M, ..., x_0 - \beta_n + M)$ look like?
Is $$ \phi^{-1}(I) = \bigcup (x_0 - \gamma_0, ..., x_n - \gamma_n ) $$ where the union is over all $\gamma$ such that $\gamma_j - \beta_i \in M$?