Nonlinear Equations - UFRJ

[SEC. 2.7: PROJECTIVE GEOMETRY 31
Denote by M D f
: H D−d1 × · · · H D−ds ↦→ H D the map
M D f : g 1 , . . . , g s ↦→ f 1 g 1 + · · · + f s g s .
Let X D be the set of all f so that Mf
D fails to be surjective. The
ideal I(X D ) is either (1) or the zero-set of the ideal of maximal subdeterminants
of Mf
D . So it is always a Zariski closed set.
By Corollary 2.7 X = ∩X D is Zariski closed.
We can use the Main Theorem of Elimination to deduce that for a
larger class of polynomial systems, the number of zeros is generically
independent of the value of the coefficients. We first will count roots
in P n .
Corollary 2.34. Let k = C. Let F be a subspace of H = H d1 × · · · ×
H dn . Let V = {(f, x) ∈ F × P n : f(x) = 0} be the solution variety.
Let π 1 : V → F and π 2 : V → P n denote the canonical projections.
Then, the critical values of π 1 are a strict Zariski closed subset of
F.
In particular, when f ∈ F is a regular value for π 1 ,
is independent of f.
n P n(f) = # ( π 2 ◦ π −1
1
)
(f)
Proof. The critical values of π 1 are the systems f ∈ F such that there
is 0 ≠ x ∈ C n+1 with
f(x) = 0 and rank(Df(x)) < n.
The rank of a n × n + 1 matrix is < n if and only if all the n × n
sub-matrices obtained by removing a column from Df(x) have zero
determinant. By Theorem 2.33, the critical values of π 1 are then the
intersection of n + 1 Zariski-closed sets, hence in a Zariski-closed set.
Because of Sard’s Theorem, the set of singular values has zero
measure. Hence, it is a strict Zariski subset of F.
Let f 0 and f 1 ∈ F be regular values of π 1 . Because Zariski open
sets are path-connected, there is a path joining f 0 and f 1 avoiding
singular values. If x 0 is a root of f 0 , then (by the implicit function
theorem) the path f t can be lifted to a path (f t , x t ) ∈ V. This implies
that f 0 and f 1 have the same number of roots in P n .