Nonlinear Equations - UFRJ

[SEC. 1.2: SHORTCOMINGS OF BÉZOUT’S THEOREM 7
Here is what happened: when n ≥ 2, no system of the form
Ax − λx = 0 can be generic in the space of polynomials systems of
degree (2, 2, · · · , 2, 1). This situation is quite common, and it pays
off to refine Bézout’s bound.
One can think of the system above as a bi-linear homogeneous
system, of degree 1 in the variables x 1 , . . . , x n−1 , x n and degree 1 in
variables λ, µ. The equations are now
µAx − λx = 0.
The eigenvectors x are elements of projective space P n and the
eigenvalue is (λ : µ) ∈ P = P 1 . Examples of “ghost” roots in P n+1
but not in P n−1 × P are, for instance, the codimension 2 subspace
λ = µ = 0.
In general, let n = n 1 + · · · + n s be a partition of n. We will
divide variables x 1 , . . . , x n into s sets, and write x = (x 1 , . . . , x s ) for
x i ∈ C ni . The same convention will hold for multi-indices.
Theorem 1.5 (Multihomogeneous Bézout). Let n = n 1 + · · · + n s ,
with n 1 , . . . , n s ∈ N. Let d ij ∈ Ϝ ≥0 be given for 1 ≤ i ≤ n and
1 ≤ j ≤ s.
Let B denote the coefficient of ω n1
1 ωn2 2 · · · ωns s in
n∏
(d i1 ω 1 + · · · + d is ω s ) .
i=1
Then, for a generic choice of coefficients f ia ∈ C, the system of
equations
∑
f 1 (x) = f 1a x a1
1 · · · xas s
|a 1|≤d 11
.
. |a s|≤d 1s
f n (x) =
∑
f na x a1
1 · · · xas s
|a 1|≤d n1
.
.
|a s|≤d ns