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38 CHAPTER 3. SOLVING POLYNOMIAL SYSTEMS OF EQUATIONS
6 monomials 1, x 1 , x 2 1, x 2 , x 2 2, and x 3 2. The matrices A T x 1
and A T x 2
are given below:
⎛
⎞
0 1 0 0 0 0
0 0 1 0 0 0
3
0 1 0 −2 0
A T 2
x 1
=
0 0 0 2 − 3 ,
2
0
⎜
0 0 0 0 2 − 3 2 ⎟
⎝
0 0 0 − 24 66
13 13
− 36 ⎠
13
⎛
0 0 0 1 0 0
0 0 0 2 0 − 3 2
9
0 0 0 4 −6
A T 4
x 2
=
0 0 0 0 1 0
⎜
0 0 0 0 0 1
⎝
16
0 0 0
13
− 44
13
124
39
⎞
.
⎟
⎠
(3.23)
Let us, for example, have a look at the entry (6, 4) of the matrix A T x 1
. This entry
denotes the coefficient of the 4 th monomial of the basis B in the normal form of the
product of x 1 and the 6 th monomial of B. The product of x 1 and the 6 th monomial
of B is x 1 x 3 2 and its normal form takes the form − 36
13 x3 2 + 66
13 x2 2 − 24
13 x 2. The coefficient
of the 4 th monomial of B in this normal form is − 24
13
which is exactly the element at
position (6, 4) in the matrix A T x 1
. It is easy to verify that the matrices commute and,
thus, it holds that A T x 1
A T x 2
= A T x 2
A T x 1
.
The eigenvalue pairs (λ 1 ,λ 2 ) of the matrices A T x 1
and A T x 2
, respectively, are the
solutions (x 1 ,x 2 ) of the system (3.22): (−1, 0), (1, 0), (0, 1.3333), (0, 0), (0.61538 +
0.39970i, 0.92308 − 0.26647i) and (0.61538 − 0.39970i, 0.92308 + 0.26647i).
However, once the eigenvalues and eigenvectors of the matrix A T x 1
are computed,
the eigenvalues of A T x 2
, which are the x 2 -coordinates of system (3.22), can also be
read off from the corresponding eigenvectors v. The eigenvectors are structured conformably
with the basis B, which means that the fourth element of v denotes the
value for the corresponding eigenvalue λ 2 (after normalization of v to make its first
element equal to 1) without having to construct the matrix A T x 2
explicitly and without
having to perform additional eigenvalue computations.
Sometimes it can be helpful to split off unwanted solutions of a system of polynomial
equations. This can be done by a so-called primary decomposition of the ideal
I = 〈f 1 ,f 2 ,f 3 〉 (see [98]). In this case however it would not be helpful because of the
small dimensions of the matrices involved. Moreover, it goes beyond the scope of this
chapter to go into all details of this. A primary decomposition of the ideal I could