An idealized strategy is an iterative procedure
which would invoke Categorize on
the pair of sets of
polynomial
equations C and 
( is initially empty),
allow creation of motivated unknowns,
allow for unknowns to be redeclared as knowns
and allow for human intervention.
Human intervention would consist
of selecting
particular equations which are nice in some way
(e.g., this is a Ricatti equation), adding
them to the collection
and possibly selecting a motivated unknown y as described
above.
This procedure would
terminate when the ideal generated by
equals the ideal generated by 
. This,
of course, is equivalent to the condition
that 
is empty for each 
.
In other words, an idealized
strategy terminates when all of the polynomial equations
which can be derived algebraically and involve
unknowns can be derived algebraically from the
equations in together with the equations
which do not involve unknowns.
If the reader would indulge us, we now discuss a more
visual way of thinking about idealized strategies.
For ease of exposition, the discussion in this paragraph will
ignore the possibility of
introducing motivated unknowns,
the redeclaration of unknowns as knowns
or human intervention.
For each pair of collections of polynomials C and
, suppose that one has
an idealized display:
When one begins using an idealized
strategy, C would be the collections of
equations coming from the problem at hand and
would be initially empty. During each iteration of a strategy,
one would choose a number of polynomials from
the ideal
generated by ,
(that is, below the
dark line) and place them in (that is, above the
dark line). The idealized display would then adjust itself
so that what appears in 
above the line and
what appears below the line corresponds to
the new value of . When there are no equations
below the dark line, the idealized strategy is complete.
If 
and, for 
,
is produced from C and
using one step
of an idealized
strategy and
each equation which is in
but not in
has at most 1 (motivated) unknown, then we say that
is derivable from C by an idealized strategy.
A question about strategies is
Which classical results can be derived using an idealized strategy?
This question gives an abstract statement of what we address in this paper.
One intends, of course, to move beyond this question to the derivation of new theorems. However, in the early stages of the subject, we think that the most urgent task is to understand classical theorems from this viewpoint. Evaluating these new and unusual techniques on a new and not well understood problem gives less of an idea of their strength than evaluating them on a well understood problem. Typically, if we derive a new theorem using symbol manipulation, then one can go back and produce a proof by hand, using ideas gotten from the symbolic manipulation.