Nonlinear Equations - UFRJ

80 [CH. 6: EXPONENTIAL SUMS AND SPARSE POLYNOMIAL SYSTEMS
This is 1/n! times the coefficient in λ 2 1λ 2 2 · · · λ 2 n of
1
π
∫M
n ω n
Note that if ω is degenerate, then the expected number of roots
is zero.
It is time for the calculus of reproducing kernels. If K(x, y) =
K(y, x) is smooth, and K(x, x) is non-zero, then we define ω K as the
form given by the formulas of Lemma 5.10:
√ −1 ∑
ω = g ij dz i ∧ d¯z j
2
with
g ij (x) =
ij
(
1
K ij (x, x) − K )
i·(x, x)K·j (x, x)
.
K(x, x)
K(x, x)
Proposition 6.12. Let A = λ 1 A 1 + · · · + λ n A n . Let
K A (x, y) = ∏ K Ai (λx, λy)
with K Ai as above. Then, K A is a reproducing kernel corresponding
to exponential sums with support in A, and
∫
∫
∫
ωK ∧n
A
= λ 1 ωK ∧n
A1
+ · · · + λ n ωK ∧n
An
M
M
M
In particular, the integral of the root density is precisely π n /n!
times the mixed volume of A 1 , . . . , A n . Since the proof of Proposition
6.12 is left to the exercises.
Now we come to the points at toric infinity.
Definition 6.13. Let A 1 , . . . , A n be polytopes in R n . A facet of
(A 1 , . . . , A n ) is a n-tuple (B 1 , . . . , B n ) such that there is one linear
form η in R n and the points of each B i are precisely the maxima of
η in A i .
Let B 1 , . . . , B n be the lattice points in facet (B 1 , . . . , B n ). A system
f has a root at (B 1 , . . . , B n ) infinity if and only if (f 1,B1 , . . . , f n,Bn )