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3.5. EXAMPLE 39
be:
I = 〈13x 2 2 − 24x 2 +12, 2x 1 +3x 2 − 4〉 +
〈x 2 , (x 1 − 1)〉 + 〈x 2 , (x 1 +1)〉 + 〈3x 2 − 4, x 1 〉 + 〈x 1 ,x 2 〉
(3.24)
of which the last four ideals are responsible for the four solutions which contain a zero
in one of its coordinates. In a situation where these solutions are meaningless, they
can be split off from the solution set by removing the corresponding four ideals and
one may continue with computing the solutions of {13x 2 2 − 24x 2 +12, 2x 1 +3x 2 − 4}.
When counting real and complex solutions of the system of polynomial equations
(3.22), one can construct the matrix M mentioned in Theorem 3.5, which has the
following structure:
⎛
⎞
6.000 1.231 2.438 3.180 3.340 3.550
1.231 2.438 −0.124 1.349 1.355 1.255
⎟ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ 2.438 −0.124 1.612 0.666 0.826 0.910
M =
. (3.25)
3.180 1.349 0.666 3.340 3.550 3.897
⎜ 3.340 1.355 0.826 3.550 3.897 4.484
⎝
3.550 1.255 0.910 3.897 4.484 5.438
The eigenvalues of M are: 16.632, 3.5083, 2.0467, 0.39722, 0.15356, and
−0.013531. The difference between the number of positive and negative eigenvalues
of the matrix M indicates that there are four real solutions and two non-real complex
solutions of the system of polynomial equations (3.22), which is consistent with our
earlier findings.