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100 CHAPTER 3. POLYNOMIAL EQUATIONS
There are seven irreducible monomials: 1, x, x 2 , x 3 , x 4 , y and xy. We know
that x satisfies a quintic, and y then satisfies ( x 2 − 2 ) y−x 3 +2 x. When x 2 = 2,
this vanishes, so our quintic for x decomposes into (x 2 − 2)(x 3 − 3), and the
whole solution reduces to
〈
x 2 − 2, y 2 − x 〉 ∪ 〈 x 3 − 3, y − x 〉 . (3.42)
Unfortunately, we do not have a convenient syntax to express this other than
via the language of ideals. We are also very liable to fall into the ‘too many
solutions’ trap, as in equation (3.8): Maple resolves the first component (in
radical form) to
{
y = 4√ 2, x = √ }
2 , (3.43)
and the second one to {
y = 3√ 3, x = 3√ }
3 , (3.44)
both of which lose the connections between x and y (x = y 2 in the first case,
x = y in the second).
We are also dependent on the choice of order, since with x > y the Gröbner
basis is
[6 − 3 y 4 − 2 y 3 + y 7 , 18 − 69 y 2 − 9 y 4 − 46 y + 23 y 5 − 2 y 6 + 73 x], (3.45)
and no simplification comes to mind, short of factoring the degree seven polynomial
in y, which of course is (y 3 − 3)(y 4 − 2), and using the choice here to
simplify the equation for x.
Maple’s RegularChains package, using the technology of section 3.4.1, produces
essentially equation (3.42) for the order y > x, and for x > y produces
[[ ( 2 y + y 3 + 4 y 2 + 2 ) x − 8 − 2 y 2 − 2 y 3 − 2 y, y 4 − 2],
[ ( 5 y + 3 + 4 y 2) x − 12 − 5 y 2 − 3 y, −3 + y 3 ]],
essentially the factored form of 3.45.
3.5 Equations and Inequalities
While it is possible to work in more general settings (real closed fields), we will
restrict our attention to solving systems over R. Consider the two equations
x 2 + y 2 = 1 (3.46)
x 2 + y 2 = −1. (3.47)
Over the complexes, there is little to choose between these two equations, both
define a one-dimensional variety. Over R, the situation is very different: (3.46)
still defines a one-dimensional variety (a circle), while (3.47) defines the empty
set, even though we have only one equation in two variables.