Categories

Suppose we are given a set V of unknown variables and a set of polynomial equations C. By the V-category of C we mean the collection of polynomial equations p = q of C such that the set of unknowns appearing in p-q is exactly V. That is, p = q is in the V-category of C if and only if p = q is in C and each element in V is a variable in p-q and each unknown in p-q belongs to V. The spreadsheet shown above has three non-empty categories: a {}-category which equals {U*U = 1, U U* =1, W*W = 1}, a {x}-category which equals {x = 0} and a {x*}-category which equals {x* = 0}. In addition to V-categories, we shall keep track of the singleton category which consists of equations of the form v=p where v is a single variable, p is a polynomial and satisfies the technical condition that v is greater than any of the terms appearing in p with respect to the monomial order (see §). In the example above, the singleton category consists of the equations x = 0 and x*= 0. These equations are very useful because they say one unknown can be solved for in terms of the others variables (and therefore eliminated). Note there is a (harmless) overlap between the singleton category and V-categories.