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 §.