Nonlinear Equations - UFRJ

[SEC. 1.1: BÉZOUT’S THEOREM 3
tions
f 1 (x) = ∑
|a|≤d 1
f 1a x a
f n (x) = ∑
.
|a|≤d n
f na x a
has exactly B = d 1 d 2 . . . d n roots x in C n . The number of isolated
roots is never more than B.
This can be restated in terms of homogeneous polynomials with
roots in projective space P n . We introduce a new variable x 0 (the
homogenizing variable) so that all monomials in the i-th equation
have the same degree. We denote by fi h the homogenization of f i ,
f h i (x 0 , . . . , x n ) = x di
0 f i
(
x1
, . . . , x )
n
x 0 x 0
Once this is done, if (x 0 , · · · , x n ) is a simultaneous root of all fi h ’s, so
is (λx 0 , · · · , λx n ) for all λ ∈ C. Therefore, we count complex ‘lines’
through the origin instead of points in C n+1 .
The space of complex lines through the origin is known as the
projective space P n . More formally, P n is the quotient of (C n+1 )≠0
by the multiplicative group C × .
A root (z 1 , . . . , z n ) ∈ C n of f corresponds to the line (λ, λz 1 , . . . ,
λz n ), also denoted by (1 : z 1 : · · · : z n ). That line is a root of f h .
Roots (z 0 : · · · : z n ) of f h are of two types: if z 0 ≠ 0, then
z corresponds to the root (z 1 /z 0 , . . . , z n /z 0 ) of f, and is said to be
finite. Otherwise, z is said to be at infinity.
We will give below a short and sketchy proof of Bézout’s theorem.
It is based on four basic facts, not all of them proved here.
The first fact is that Zariski open sets are path-connected. Suppose
that V is a Zariski closed set, and that y 1 ≠ y 2 are not points of
V . (This already implies V ≠ C n ). We claim that there is a path