Nonlinear Equations - UFRJ

[SEC. 5.4: AFFINE AND MULTI-HOMOGENEOUS SETTING 63
5.4 Affine and multi-homogeneous setting
We start by particularizing Theorem 5.11 for the Bézout Theorem
setting.
The space P di of all polynomials of degree ≤ d i is endowed with
the Weyl inner product [85] given by
⎧ ( ) −1
⎨ di
〈x a , x b 〉 =
if a = b
a
(5.2)
⎩
0 otherwise.
With this choice, P di is a non-degenerate fewspace with Kernel
K(x, y) = ∑ )
x a ȳ a = (1 + 〈x, y〉) di
a
|a|≤d i
(
di
The geometric reason behind Weyl’s inner product will be explained
in the next section. A consequence of this choice is that the metric
depends linearly in d i .
We compute K j·(x, x) = d j ¯x j K(x, x)/R 2 and
K jk (x, x) = δ jk d i K(x, x)/R 2 + d i (d i−1 )¯x j x k /R 4 ,
with R 2 = 1 + ‖x‖ 2 . Lemma 5.10 implies
g jk = d i
( 1
R 2 (
δ jk − ¯x jx k
R 2 ) ) ,
with R 2 = 1 + ‖x‖ 2 . Thus, if ω i is the metric form of P di
metric form of P 1 ,
and ω 0 the
n∧ n∏ n∧
ω 1 = ( d i ) ω 0 .
i=1
i=1
Comparing the bounds in Theorem 5.11 for the linear case (degree
1 for all equations) and for d, we obtain:
Corollary 5.12. Let f ∈ P d = P d1 × · · · × P dn be a zero average,
unit variance variable. Then,
i=1
E(n C n(f)) = ∏ d i