In Section 4.3 we considered the simplification of polynomials with symbols for algebraic numbers that are defined as the solutions of polynomial equations. In Section 6.2 we considered the simplification of multivariate polynomials with respect to a single polynomial side relation. In this chapter we again consider the polynomial simplification problem but this time allow for several multivariate polynomial side relations that may have symbols in common. Because of this, the supporting mathematical theory is more involved than the material in Sections 4.3 and 6.2. The algorithm in this chapter is a generalization of the algorithms in these earlier sections.

In Section 8.1 we describe the division process that is used in the simplification algorithm. In Section 8.2 we give a precise definition of the simplification problem and introduce the concept of a Gr bner basis which plays a key role in our algorithm. Finally, in Section 8.3, we give an algorithm that finds a Gr bner basis and the polynomial simplification algorithm.

8.1 Multiple Division and Reduction

To simplify the presentation, we assume that all polynomials are in the domain Q[ x ₁, , x _p]. We begin with some examples.

Example 8.1

Suppose that w, x, y, and z, satisfy the two side relations

and consider the simplification of the polynomial

with respect to the two side relations. Although it is evident from the unexpanded form in Equation (8.3) that u simplifies to 0 at all points that satisfy the side relations, this...