and unknowns
.
The first operation which we consider is called
Categorize.
There are two key ideas behind the functioning of
Categorize. Firstly, it finds equations not involving any
unknowns,
equations involving one unknown,
equations involving two unknowns,
etc.
We will give an example below which shows how
this can be beneficial.
Secondly, Categorize
applies to
polynomials
which are members of
, but which are not members of
.
More precisely, Categorize associates to two
collections of polynomials C and 
(in 
),
the collection 
of subsets
of . Each
consists of polynomials
which depend on exactly j unknowns and
which lie in 
, that is, which are members of
but
which are not members of .
Categorize is not implementable on a computer,
because subsumed in the Categorize command
is the ability to eliminate unknowns (whenever
algebraically
possible) from equations
in the original set of polynomial equations.
In fact, the paper
[TMora] shows that there does not exist a
computer algorithm which can determine whether or not
is the empty set.
For an example of how Categorize can be useful,
suppose that C is a collection of polynomial
equations in knowns 
and unknowns 
. If it could
be shown algebraically that the
Ricatti equation 
follows from C, then this Ricatti equation
would be a member of .
Knowing that 
satisfies a Ricatti equation
can be of great value since
Ricatti equation can be quickly solved numerically.
In this example,
is the empty set.
A slightly more complicated example would be if
it could be shown algebraically that
an expression, such as 
,
solved a Ricatti equation, e.g., if
follows from a collection of polynomial equations C.
The left hand side of (1.1)
would depend on three
unknowns , 
and 
and,
therefore, would be a member of 
, not .
It is natural, however, to view
(1.1) as an equation in one new
unknown y and to rewrite the left hand side of
(1.1) as the composition
where 
and 
.
In this example, we would call y a motivated unknown.
The second of our two
idealized operations is called Decompose and
associates to a polynomial p all non-trivial maximal
compositions
Decompose, therefore,
produces motivated unknowns. Decompose
will not be used, or discussed further, until
§.
