Here, we prove a theorem very similar to that of the previous chapter .
The proof is shorter but involves more advanced mathematics. It has the advantage that it does not assume the same minimality hypotheses .
If
F
(
z
)
=
G
(
z
)
H
(
z
)
{\displaystyle F(z)={\frac {G(z)}{H(z)}}}
V
{\displaystyle {\mathcal {V}}}
V
{\displaystyle {\mathcal {V}}}
h
r
=
c
−
ϵ
{\displaystyle h_{\textbf {r}}=c-\epsilon }
W
{\displaystyle W}
V
{\displaystyle {\mathcal {V}}}
c
{\displaystyle c}
C
=
∑
z
∈
W
C
∗
(
z
)
{\displaystyle C=\sum _{z\in W}C_{*}(z)}
H
d
(
M
,
M
c
−
ϵ
)
{\displaystyle H_{d}({\mathcal {M}},{\mathcal {M}}^{c-\epsilon })}
N
{\displaystyle {\mathcal {N}}}
r
{\displaystyle {\textbf {r}}}
[1]
a
n
r
∼
∑
z
∈
W
w
−
n
r
(
2
π
)
(
1
−
d
)
/
2
det
H
G
(
w
)
w
k
H
k
(
w
)
(
n
r
k
)
(
1
−
d
)
/
2
{\displaystyle a_{n{\textbf {r}}}\sim \sum _{z\in W}{\textbf {w}}^{-n{\textbf {r}}}{\frac {(2\pi )^{(1-d)/2}}{\sqrt {\det {\mathcal {H}}}}}{\frac {G({\textbf {w}})}{w_{k}H_{k}({\textbf {w}})}}(nr_{k})^{(1-d)/2}}
Let
B
{\displaystyle B}
B
′
{\displaystyle B'}
a
m
o
e
b
a
(
H
)
c
{\displaystyle amoeba(H)^{c}}
h
r
{\displaystyle h_{r}}
B
′
{\displaystyle B'}
T
=
T
e
(
x
)
{\displaystyle T=T_{e}(x)}
x
∈
B
{\displaystyle x\in B}
T
′
=
T
e
(
x
′
)
{\displaystyle T'=T_{e}(x')}
x
′
∈
B
′
{\displaystyle x'\in B'}
[2]
The intersection class
I
N
T
(
T
,
T
′
)
{\displaystyle INT(T,T')}
V
{\displaystyle {\mathcal {V}}}
T
{\displaystyle T}
T
′
{\displaystyle T'}
V
{\displaystyle {\mathcal {V}}}
transversely .[3]
If we choose the homotopy such that its time
t
{\displaystyle t}
t
{\displaystyle t}
w
{\displaystyle {\textbf {w}}}
V
{\displaystyle {\mathcal {V}}}
I
N
T
(
T
,
T
′
)
{\displaystyle INT(T,T')}
γ
{\displaystyle \gamma }
V
{\displaystyle {\mathcal {V}}}
w
{\displaystyle {\textbf {w}}}
[4]
By the Cauchy coefficient formula and residue theorem:
a
r
=
1
(
2
π
i
)
d
∫
T
F
(
z
)
z
−
r
−
1
d
z
=
1
(
2
π
i
)
d
−
1
∫
γ
R
e
s
(
F
(
z
)
z
−
r
−
1
d
z
)
+
1
(
2
π
i
)
d
−
1
∫
T
′
F
(
z
)
z
−
r
−
1
d
z
=
1
(
2
π
i
)
d
−
1
∫
γ
R
e
s
(
F
(
z
)
z
−
r
−
1
d
z
)
{\displaystyle {\begin{aligned}a_{r}&={\frac {1}{(2\pi i)^{d}}}\int _{T}F(z)z^{-r-1}dz\\&={\frac {1}{(2\pi i)^{d-1}}}\int _{\gamma }Res(F(z)z^{-r-1}dz)+{\frac {1}{(2\pi i)^{d-1}}}\int _{T}'F(z)z^{-r-1}dz\\&={\frac {1}{(2\pi i)^{d-1}}}\int _{\gamma }Res(F(z)z^{-r-1}dz)\end{aligned}}}
As a result of ...[5]
a
r
=
e
−
h
r
(
w
)
(
2
π
i
)
d
−
1
∫
γ
e
−
r
d
ϕ
(
z
)
P
(
z
)
Q
(
z
)
∏
j
=
1
d
z
j
d
z
∘
{\displaystyle a_{r}={\frac {e^{-h_{r}(w)}}{(2\pi i)^{d-1}}}\int _{\gamma }e^{-r_{d}\phi (z)}{\frac {P(z)}{Q(z)\prod _{j=1}^{d}z_{j}}}dz^{\circ }}
By theorem 5.3 and the change of variables
z
j
=
w
j
e
i
θ
j
{\displaystyle z_{j}=w_{j}e^{i\theta _{j}}}
[6]
a
n
r
∼
∑
z
∈
W
w
−
n
r
(
2
π
)
(
1
−
d
)
/
2
det
H
G
(
w
)
w
k
H
k
(
w
)
(
n
r
k
)
(
1
−
d
)
/
2
{\displaystyle a_{n{\textbf {r}}}\sim \sum _{z\in W}{\textbf {w}}^{-n{\textbf {r}}}{\frac {(2\pi )^{(1-d)/2}}{\sqrt {\det {\mathcal {H}}}}}{\frac {G({\textbf {w}})}{w_{k}H_{k}({\textbf {w}})}}(nr_{k})^{(1-d)/2}}
↑ Pemantle and Wilson 2013, pp. 174-175.
↑ Pemantle, Wilson and Melczer 2024, pp. 275.
↑ Pemantle, Wilson and Melczer 2024, pp. 276.
↑ Pemantle, Wilson and Melczer 2024, pp. 276.
↑ Pemantle, Wilson and Melczer 2024, pp. 276.
↑ Pemantle, Wilson and Melczer 2024, pp. 276.
