Nonlinear Equations - UFRJ

68 [CH. 5: REPRODUCING KERNEL SPACES
Remark 5.19. Given f in a fewspace F of equations, define E f =
{(x, f(x)) : x ∈ M}. Then E f is invariant by H × C × -action. Therefore
(E f /(H × C × , M/H, π, C)
is a line bundle. In this sense, solving a system of polynomial equations
is the same as finding simultaneous zeros of n line bundles.
Theorem 5.20 (Homogeneous root density). Let K be a locally measurable
set of M/H. Let F 1 , . . . , F n be fewspaces on the quotient
M/H, with ω 1 , . . . , ω n be the induced (possibly degenerate) symplectic
forms. Assume that f = f 1 , . . . , f n is a zero average, unit variance
variable in F = F 1 × · · · × F n . Then,
E(n K (f)) = 1
π
∫K
n ω 1 ∧ · · · ∧ ω n .
Proof. There is a covering U α of M/H such that each U α may be
diffeomorphically embedded in M Now, the F i are fewspaces of functions
in U α .
Write K as a disjoint union of sets K α where each K α is measurable
and contained in U α . By Theorem 5.11,
E(n Kα (f)) = 1 ∫
π n ω 1 ∧ · · · ∧ ω n .
K α
Then we add over all the α’s.
It is time to explain the choice of the inner product (5.2) and
(5.3). Suppose that we want to write f ∈ H d as a symmetric tensor.
Then,
∑
f(x) =
T j1,...,j d
x j1 x j2 · · · x jid
with
1≤x j1 ,...,x jd ≤n
1
T j1,...,j d
= (
)f ej1 +···+e
d
jd
.
e j1 + · · · + e jd
The Frobenius norm of T is precisely ‖T ‖ F = ‖f‖. The reader
shall check (Exercise 5.3) that ‖T ‖ F is invariant for the U(n + 1)-
action on C n+1 .