Nonlinear Equations - UFRJ

[SEC. 5.5: COMPACTIFICATIONS 65
This is a polynomial in variables Z 1 = ω 11 , . . . , Z s = ω ss . Notice
that Z 1 ∧Z 2 = Z 2 ∧Z 1 so we may drop the wedge notation. Moreover,
Z ni+1
i = 0. Hence, only the monomial in Z n1
1 Zn2 2 · · · Zns s may be
nonzero.
Corollary 5.14. Let B be the coefficient of Z n1
1 Zn2 2 · · · Zns s
∏
(di1 Z 1 + · · · + d is Z s ).
in
Let f ∈ F = F 1 × · · · × F n be a zero average, unit variance variable.
Then,
E(n C n(f)) = B
Proof. By Theorem 5.11,
E(n C n(f)) = 1 ∫
π n
= B π n ∫ Cn ∧
ωi
K
ω 11 ∧ · · · ∧ ω
} {{ 11 ∧ · · · ∧ ω
}
s1 ∧ · · · ∧ ω
} {{ s1
}
n 1times
n stimes
In order to evaluate the right-hand-term, let G j be the space of
affine polynomials on the j-th set of variables. Its associated symplectic
form is ω i1 .
A generic polynomial system in
G = G 1 × · · · G
} {{ } 1 × · · · × G s × · · · G
} {{ } s
n 1times
n stimes
is just a set of decoupled linear systems, hence has one root. Hence,
1 = 1 ∫
π n ω 11 ∧ · · · ∧ ω 11 ∧ · · · ∧ ω
C
} {{ }
s1 ∧ · · · ∧ ω
} {{ s1
}
n
n 1times
n stimes
and the expected number of roots of a multi-homogeneous system
is B.
5.5 Compactifications
The Corollaries in the section above allow to prove Bézout and Multi-
Homogeneous Bézout theorems, if one argues as in Chapter 1 that