Varieties in $(\mathbf{P}^1(\bar{\mathbf{F}}))^n$ by Elimination and Extension

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.