Algebraic characterization of commutative rings with Krull dimension=1,2, or 3 A commutative ring $R$ (with $1$) is $0$-dimensional if and only if $R/\sqrt 0$ is von Neumann regular. Besides this result, there is a wealth of information about zero-dimensional rings.
I could not find any information about rings of higher Krull dimension. If the problem for general commutative rings is too hard, how about the case when $R$ is Noetherian and/or an integral domain?

What about the following result due to Coquand and Lombardi
(A short proof for the Krull dimension of a polynomial ring.
Amer. Math. Monthly 112 (2005), no. 9, 826–829.) 
Let $R$ be a commutative unital ring. For $N\in \mathbb N$, 
the following  assertions are equivalent:
(1)The Krull dimension of $R$ is at most $N$.
(2) For all $(a_0,\dots, a_{N})\in R^{N+1}$ there exists  $(x_0,\dots, x_{N})\in R^{N+1}$ and $(n_0,\dots, n_{N})\in \mathbb N^{N+1}$ such that
$$
 a_0^{n_0}\Biggl( a_1^{n_1} \Bigl(\cdots \bigl( a_{N}^{n_{N}}(1 +a_{N} x_{N})+\cdots\bigr)\Bigr) +a_0x_0\Biggr)=0.
$$
