Nonlinear Equations - UFRJ

[SEC. 2.1: SYLVESTER’S RESULTANT 13
zero. The polynomials f and g have a common root if and only if the
linear map M f,g : P e−1 × P d−1 → P d+e−1 defined by
is degenerate.
a, b ↦→ af + bg
If we assimilate each P d to k d+1 by associating each a(x) = a d x d +
· · · + a 0 to [a d , · · · , a 0 ] T ∈ k d+1 , the linear map M f,g corresponds to
the Sylvester matrix
⎡
⎤
f d
g e
. f d−1 f d g ..
e−1 . f .. .
d−1 ge−2
.. ge
. .
.. .
fd
.. ge−1
Syl(f, g) =
. f d−1 . g e−2
.
f 1
f 0 f 1 . g 1 .
. f .. . 0 g0 ..
⎢
⎣
. .. . ⎥
f1
..
g1
⎦
f 0 g 0
The Sylvester resultant is usually defined as
Res x (f(x), g(x)) def
= det Syl(f, g).
Proof of Theorem 2.1. Assume that z ∈ k is a common root for f
and g. Then,
[z d+e z d+e−1 · · · z 1] Syl(f, g) = a(z)f(z) + b(z)g(z) = 0.
Therefore the determinant of Syl(f, g) must vanish. Hence M f,g is
degenerate.
Reciprocally, assume that M f,g is degenerate. Then there are
a ∈ P e−1 , b ∈ P d−1 so that af + bg ≡ 0. Assume for simplicity that
d ≤ e and g e ≠ 0. By the Fundamental Theorem of Algebra, g admits
e roots z 1 , . . . , z e (counted with multiplicity). By the pigeonhole