TABLE OF CONTENTS Most symbolic computation involves mathematical expressions that contain more than one symbol or generalized variable. In order to manipulate these expressions, we must extend the concepts and algorithms described in Chapter 4 to polynomials with several variables. This generalization is the subject of this chapter. It includes a description of coefficient domains and a recursive view of multivariate polynomials (Section 6.1), versions of polynomial division and expansion (Section 6.2), greatest common divisor algorithms (Section 6.3), and the extended Euclidean algorithm (Section 6.3).
A multivariate polynomial u in the set of distinct symbols { x 1, x 2, , x p} is a finite sum with (one or more) monomial terms of the form
where the coefficient c is in a coefficient domain K and the exponents n j are non-negative integers. (The axioms for K (which may not be a field) are given in Definition 6.2 below.) The notation K[ x 1, x 2, , x p] represents the set of polynomials in the symbols x 1, x 2, , x p with coefficients in K. For example, Z[ x, y] represents all polynomials in x and y with coefficients that are integers.
A particularly important instance when K is not a field has to do with the recursive representation of multivariate polynomials. For example, let Q[ x,
