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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.

*Helton *

Wed Jul 3 10:27:42 PDT 1996