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34 CHAPTER 3. SOLVING POLYNOMIAL SYSTEMS OF EQUATIONS
variable x. Solving for x yields the solutions:
x =2,y =1,z = −1
x =0,y =0,z =0
x =1,y =1,z =0
x =5+3 √ 2,y =2+ √ 2,z =1+ √ 2
x =5− 3 √ 2,y =2− √ 2,z =1− √ 2
(3.17)
An immediate application of a Gröbner basis to solve polynomial equations in C
is derived from the Weak Hilbert Nullstellensatz (Theorem 1 in Chapter 4 of [28])
and can be formulated as follows:
Theorem 3.2. (Consistency) If I = 〈f 1 ,...,f m 〉 is an ideal in C[x 1 ,...,x n ], then
the system of polynomial equations f 1 (x 1 ,...,x n ) = ... = f m (x 1 ,...,x n ) = 0 is
unsolvable if and only if the reduced Gröbner basis of I is {1}. Thus the variety
V (I) =∅.
Proof. The proof of this theorem is given in [23].
□
Example 3.5. Consider the following system of polynomial equations:
⎧
f 1 = x 2 y − z 3 =0
⎪⎨
⎪⎩
f 2 = 2xy − 4z − 1 = 0
f 3 = y 2 − z =0
f 4 = x 3 − 4y =0
(3.18)
The reduced Gröbner basis for I = 〈f 1 ,f 2 ,f 3 ,f 4 〉 with respect to the lex ordering is
{1}, hence the system is unsolvable.
A similar theorem in terms of the Gröbner basis of the corresponding ideal exists
for a system with finitely many solutions. This theorem is based on Theorem 6 of
Chapter 5 of [28]:
Theorem 3.3. (Finiteness) If I = 〈f 1 ,...,f m 〉 is an ideal in C[x 1 ,...,x n ], then
the system of polynomial equations f 1 (x 1 ,...,x n )=... = f m (x 1 ,...,x n )=0has
finitely many solutions if and only if for all i, 1 ≤ i ≤ n, there exists a power of x i
which belongs to the ideal generated by the leading terms of I: 〈LT (I)〉
Proof. The proof of this theorem is given in [23] and in [28].
□
An algebraic method which uses normal form computations in a quotient algebra
to solve zero-dimensional ideals of multivariate polynomials, is presented in the next
section.