Nonlinear Equations - UFRJ

64 [CH. 5: REPRODUCING KERNEL SPACES
Remark 5.13. Mario Wschebor pointed out that if one could give a
similar expression for the variance (which is zero) it would be possible
to deduce and ‘almost everywhere’ Bézout’s theorem from a purely
probabilistic argument.
Now, let F i is the space of polynomials with degree d ij in the j-th
set of variables. We write x = (x 1 , . . . , x s ) for x i ∈ C ni , and the
same convention holds for multi-indices.
The inner product will be defined by:
〈x a1
1 . . . xan s , x b1
1 . . . xbn
The integral kernel is now
s 〉 = δ a 1b 1 · · · δ
( ) asb s
di1
· · ·
a 1
K(x, y) = (1 + 〈x 1 , y 1 〉) di1 · · · (1 + 〈x s , y s 〉) dis
(
dis
a s
) (5.3)
We need more notations: the j-th variable belongs to the l(j)-th
group, and R 2 l = 1 + ‖x l‖ 2 .
With this notations,
¯x j K(x, x)
K j·(x, x) = d l(j)
Rl(j)
2
K(x, x)
K jk (x, x) = δ jk d l(j)
Rl(j)
2 + d l(j) (d l(k) − δ l(j)l(k) )
(
)
δ jk
¯x j x k
g jk = d l(j) − δ l(j)l(k)
R 2 l(j)
R 2 l(j) R2 l(k)
¯x j x k
R 2 l(j) R2 l(k)
Recall that ω i is the symplectic form associated to F i . We denote
by ω jd the form associated to the polynomials that have degree ≤ d in
the j-th group of variables, and are independent of the other variables.
From the calculations above,
ω i = ω 1d1 + · · · + ω sds = d i1 ω 11 + · · · + d is ω s1
Hence, ∧
ωi = ∧ d i1 ω 11 + · · · + d is ω s1 .