  This paper contains a theory of elimination and extension to compute
varieties symbolically, based on using {\em coordinates} from
$(\mathbf{P}^1(\bar{\mathbf{F}}))^n$ and disjoint {\em parts} of varieties
(defined by both equality and inequality constraints), leading to a recursive
algorithm to compute said varieties by extension at the level of {\em parts} of
a variety. {\sc Macaulay2} code for this is included along with an example.
This is a first step in the author's project of giving a purely algebraic
theory of desingularization of function fields, in that that project relies
heavily on using this type of coordinates for function field elements and on
partitioning a set of valuations into disjoint sets similarly.
