Maximal ideals in univariate polynomial rings $R[X]$ have a nice characterization in that they all are of the form $(E)$, for some irreducible $E\in R[X]$. This allows for a systematic way to construct maximal ideals in this setting.
I'm looking to do the same for multivariate polynomial rings. Let $k$ be a field (not algebraically closed -- imagine that it's $\mathbb{F}_p$ for some prime $p$), and let $k[x_1,\ldots,x_n]$ be its polynomial ring in $n$ variables.
More specifically, I'm looking for maximal ideals $I\subseteq k[x_1,\ldots,x_n]$ such that for any polynomial $f\in k[x_1,\ldots,x_n]$ of total degree at most $d$, $f$ is the unique degree $\leq d$ polynomial such that $f = f \mod I$. Otherwise, there exists a degree $\leq d$ polynomial $g$ such that $g = f \mod I$. Intuitively, I would like to have something that behaves like modding by an irreducible $E$ in a univariate polynomial ring: if $E$ is of degree $d+1$, then $f \mod E$ has the behavior I described.
Any other pointers to examples/characterizations of maximal ideals in multivariate polynomial rings would be appreciated!
Thank you!