Rankings of Partial Derivatives and Elimination Algorithms for PDE Colin Rust Mathematics Department University of Chicago Abstract Let $m,n$ be positive integers, $\N = \{0,1,2, ... \}$ and $\N_n = \{ 1,\ldots ,n\}$. A ranking $\leq$ is a total order of $\N^m\times \N_n$ such that $(a,i) \leq (b,j)$ implies $(a+c,i)\leq (b+c,j)$ for $a$, $b$, $c$ $\in \N^m$ and $i,j$ $\in \N_n$. We describe an approach to such rankings which yields a theorem which can describe any such ranking by finite explicit real data. The case $n=1$ corresponds to term-orderings of monomials which are crucial inputs for Buchberger's Gr\"obner Basis algorithm for polynomial rings. The case $n>1$ corresponds to rankings of partial derivatives which are inputs in algorithms in differential algebra and Buchberger's algorithm for free modules over polynomial rings. A subclass of such rankings determined by finite integer data is given which is sufficient for effective implementation of such rankings. This has been implemented in the symbolic language Maple. The rankings considered by Riquier are a special case of those considered here. Examples including applications to initial value problems and differential elimination algorithms, such as Riquier's algorithm are given.