3 Elimination Theory This chapter will study systematic methods for eliminating variables from systems of polynomial equations. The basic strategy of elimination theory will be given in two main theorems: the Elimination Theorem and the Extension Theorem. We will prove these results using Groebner bases and the classic theory of resultants. The geometric interpretation of elimination will also be explored when we discuss the Closure The- orem. Of the many applications of elimination theory, we will treat two in detail: the implicitization problem and the envelope of a family of curves. §1 The Elimination and Extension Theorems To get a sense of how elimination works, let us look at an example similar to those discussed at the end of Chapter 2. We will solve the system of equations x 2 + y + z = 1, x + y 2 + z = 1, (1) x + y + z 2 = 1. If we let I be the ideal (2) I =〈x 2 + y + z − 1, x + y 2 + z − 1, x + y + z 2 − 1〉, then a Groebner basis for I with respect to lex order is given by the four polynomials (3) g 1 = x + y + z 2 − 1, g 2 = y 2 − y − z 2 + z , g 3 = 2 yz 2 + z 4 − z 2 , g 4 = z 6 − 4z 4 + 4z 3 − z 2 . It follows that equations (1) and (3) have the same solutions. However, since g 4 = z 6 − 4z 4 + 4z 3 − z 2 = z 2 (z − 1) 2 (z 2 + 2z − 1) involves only z , we see that the possible z ’s are 0,1 and −1 ± √ 2. Substituting these values into g 2 = y 2 − y − z 2 + z = 0 and g 3 = 2 yz 2 + z 4 − z 2 = 0, we can determine 115