We attach a few conditions to the critical point
w
{\displaystyle {\textbf {w}}}
and the function
H
{\displaystyle H}
of this chapter. Later chapters relax or modify these conditions.
H
{\displaystyle H}
can be factored into
H
1
H
2
⋯
H
n
{\displaystyle H_{1}H_{2}\cdots H_{n}}
where no factor has a power and no factor is repeated.[1]
The critical point
w
{\displaystyle {\textbf {w}}}
is one of[2] [3]
Strictly minimal : the only critical point on its polytorus, i.e.
V
∩
T
(
w
)
=
{
w
}
{\displaystyle {\mathcal {V}}\cap T({\textbf {w}})=\{{\textbf {w}}\}}
Finitely minimal : one of a finite number of critical point on its polytorus, i.e.
1
<
#
(
V
∩
T
(
w
)
)
<
∞
{\displaystyle 1<\#({\mathcal {V}}\cap T({\textbf {w}}))<\infty }
Torally minimal : one of an infinite number of critical point on its polytorus, i.e.
#
(
V
∩
T
(
w
)
)
=
∞
{\displaystyle \#({\mathcal {V}}\cap T({\textbf {w}}))=\infty }
and the the torality hypothesis is satisfied...
At least one of the partial derivatives
∂
H
∂
w
k
(
w
)
≠
0
{\displaystyle {\frac {\partial H}{\partial w_{k}}}({\textbf {w}})\neq 0}
.[4]
Formally, it means the gradient map of
H
{\displaystyle H}
at
w
{\displaystyle {\textbf {w}}}
is not equal to the zero vector.
∇
H
(
w
)
≠
0
{\displaystyle \nabla H({\textbf {w}})\neq {\textbf {0}}}
where:
∇
f
(
p
)
=
[
∂
f
∂
x
1
(
p
)
⋮
∂
f
∂
x
n
(
p
)
]
{\displaystyle \nabla f(p)={\begin{bmatrix}{\frac {\partial f}{\partial x_{1}}}(p)\\\vdots \\{\frac {\partial f}{\partial x_{n}}}(p)\end{bmatrix}}}
Quadratically nondegenerate [ edit | edit source ]
In the proof, we make use of the Hessian matrix
H
{\displaystyle {\mathcal {H}}}
[5]
H
=
[
∂
2
ϕ
(
w
)
∂
z
i
∂
z
j
]
{\displaystyle {\mathcal {H}}=\left[{\frac {\partial ^{2}\phi ({\textbf {w}})}{\partial z_{i}\partial z_{j}}}\right]}
where
ϕ
{\displaystyle \phi }
is a function we will see later.
Quadratic nondegeneracy means the Hessian matrix is nonsingular, a fact we use in the proof.
One quadratically nondegenerate smooth point [ edit | edit source ]
Let
F
(
z
)
=
G
(
z
)
H
(
z
)
{\displaystyle F({\textbf {z}})={\frac {G({\textbf {z}})}{H({\textbf {z}})}}}
where
H
(
z
)
{\displaystyle H(z)}
is squarefree and has a strictly minimal , smoothly varying and quadratically nondegenerate critical point
w
=
(
w
1
,
⋯
,
w
d
)
∈
C
∗
d
{\displaystyle {\textbf {w}}=(w_{1},\cdots ,w_{d})\in \mathbb {C} _{*}^{d}}
in the direction
r
=
(
r
1
,
⋯
,
r
d
)
∈
N
∗
d
{\displaystyle {\textbf {r}}=(r_{1},\cdots ,r_{d})\in \mathbb {N} _{*}^{d}}
. Then,
a
n
r
∼
w
−
n
r
(
2
π
)
(
1
−
d
)
/
2
det
H
G
(
w
)
w
k
∂
H
∂
w
k
(
w
)
(
n
r
k
)
(
1
−
d
)
/
2
{\displaystyle a_{n{\textbf {r}}}\sim {\textbf {w}}^{-n{\textbf {r}}}{\frac {(2\pi )^{(1-d)/2}}{\sqrt {\det {\mathcal {H}}}}}{\frac {G({\textbf {w}})}{w_{k}{\frac {\partial H}{\partial w_{k}}}({\textbf {w}})}}(nr_{k})^{(1-d)/2}}
where
H
{\displaystyle {\mathcal {H}}}
is the Hessian matrix of
ϕ
(
θ
)
{\displaystyle \phi (\theta )}
at
w
{\displaystyle {\textbf {w}}}
.[6] [7]
By the multivariate Cauchy formula [8] [9]
a
n
r
=
1
(
2
π
i
)
d
∫
T
F
(
z
)
d
z
z
n
r
+
1
{\displaystyle a_{n{\textbf {r}}}={\frac {1}{(2\pi i)^{d}}}\int _{T}F({\textbf {z}}){\frac {d{\textbf {z}}}{{\textbf {z}}^{n{\textbf {r}}+1}}}}
To make use of the implicit function theorem , we choose one coordinate from the critical point for which
∂
H
∂
w
k
(
w
)
≠
0
{\displaystyle {\frac {\partial H}{\partial w_{k}}}({\textbf {w}})\neq 0}
, possible because it is smooth . Call this coordinate
w
k
{\displaystyle w_{k}}
. Call the projection of the critical point
w
{\displaystyle {\textbf {w}}}
into
d
−
1
{\displaystyle d-1}
coordinates by
w
∘
{\displaystyle {\textbf {w}}^{\circ }}
and the projection of the direction
r
{\displaystyle {\textbf {r}}}
into
d
−
1
{\displaystyle d-1}
coordinates by
r
∘
{\displaystyle {\textbf {r}}^{\circ }}
.[10]
Rewrite the above Cauchy formula as an iterated integral , defining
T
∘
{\displaystyle T^{\circ }}
as the torus through
w
∘
{\displaystyle {\textbf {w}}^{\circ }}
[11] [12]
a
n
r
=
1
(
2
π
i
)
d
∫
T
∘
(
w
∘
)
−
n
r
∘
(
∫
|
w
k
|
=
ρ
−
δ
s
w
k
−
n
r
k
F
(
w
∘
,
w
k
)
d
w
k
w
k
)
d
w
∘
w
∘
{\displaystyle a_{n{\textbf {r}}}={\frac {1}{(2\pi i)^{d}}}\int _{T^{\circ }}({{\textbf {w}}^{\circ }})^{-n{\textbf {r}}^{\circ }}\left(\int _{|w_{k}|=\rho -\delta _{s}}w_{k}^{-nr_{k}}F({\textbf {w}}^{\circ },w_{k}){\frac {dw_{k}}{w_{k}}}\right){\frac {d{\textbf {w}}^{\circ }}{{\textbf {w}}^{\circ }}}}
By the implicit function theorem, there exists a neighbourhood
N
{\displaystyle {\mathcal {N}}}
of
w
∘
{\displaystyle {\textbf {w}}^{\circ }}
such that
H
(
w
∘
,
z
)
=
0
⟺
z
=
g
(
w
∘
)
{\displaystyle H({\textbf {w}}^{\circ },z)=0\iff z=g({\textbf {w}}^{\circ })}
[13] [14]
I
=
1
(
2
π
i
)
d
∫
N
(
w
∘
)
−
n
r
∘
(
∫
|
w
k
|
=
ρ
−
δ
s
w
k
−
n
r
k
F
(
w
∘
,
w
k
)
d
w
k
w
k
)
d
w
∘
w
∘
{\displaystyle I={\frac {1}{(2\pi i)^{d}}}\int _{\mathcal {N}}({{\textbf {w}}^{\circ }})^{-n{\textbf {r}}^{\circ }}\left(\int _{|w_{k}|=\rho -\delta _{s}}w_{k}^{-nr_{k}}F({\textbf {w}}^{\circ },w_{k}){\frac {dw_{k}}{w_{k}}}\right){\frac {d{\textbf {w}}^{\circ }}{{\textbf {w}}^{\circ }}}}
and
I
′
=
1
(
2
π
i
)
d
∫
N
(
w
∘
)
−
n
r
∘
(
∫
|
w
k
|
=
ρ
+
δ
s
w
k
−
n
r
k
F
(
w
∘
,
w
k
)
d
w
k
w
k
)
d
w
∘
w
∘
{\displaystyle I'={\frac {1}{(2\pi i)^{d}}}\int _{\mathcal {N}}({\textbf {w}}^{\circ })^{-n{\textbf {r}}^{\circ }}\left(\int _{|w_{k}|=\rho +\delta _{s}}w_{k}^{-nr_{k}}F({\textbf {w}}^{\circ },w_{k}){\frac {dw_{k}}{w_{k}}}\right){\frac {d{\textbf {w}}^{\circ }}{{\textbf {w}}^{\circ }}}}
As a result of the implicit function theorem, for any fixed
w
∘
{\displaystyle {\textbf {w}}^{\circ }}
there is a unique pole a
z
=
g
(
w
∘
)
{\displaystyle z=g({\textbf {w}}^{\circ })}
inside the annulus.[15]
The difference between the two inner integrals of
I
{\displaystyle I}
and
I
′
{\displaystyle I'}
is that the latter has this pole inside and therefore by integration with residues [16]
∫
|
w
k
|
=
ρ
+
δ
s
w
k
−
n
r
k
F
(
w
∘
,
w
k
)
d
w
k
w
k
−
∫
|
w
k
|
=
ρ
−
δ
s
w
k
−
n
r
k
F
(
w
∘
,
w
k
)
d
w
k
w
k
=
(
2
π
i
)
R
e
s
(
g
(
w
∘
)
−
n
r
k
F
(
w
∘
,
w
k
)
w
k
;
w
k
=
g
(
w
∘
)
)
{\displaystyle \int _{|w_{k}|=\rho +\delta _{s}}w_{k}^{-nr_{k}}F({\textbf {w}}^{\circ },w_{k}){\frac {dw_{k}}{w_{k}}}-\int _{|w_{k}|=\rho -\delta _{s}}w_{k}^{-nr_{k}}F({\textbf {w}}^{\circ },w_{k}){\frac {dw_{k}}{w_{k}}}=(2\pi i)Res\left(g({\textbf {w}}^{\circ })^{-nr_{k}}{\frac {F({\textbf {w}}^{\circ },w_{k})}{w_{k}}};w_{k}=g({\textbf {w}}^{\circ })\right)}
therefore[17]
χ
=
I
−
I
′
=
1
(
2
π
i
)
d
−
1
∫
N
(
w
∘
)
−
n
r
∘
g
(
w
∘
)
−
n
r
k
ψ
(
w
∘
)
d
w
∘
w
∘
{\displaystyle \chi =I-I'={\frac {1}{(2\pi i)^{d-1}}}\int _{\mathcal {N}}({{\textbf {w}}^{\circ }})^{-n{\textbf {r}}^{\circ }}g({\textbf {w}}^{\circ })^{-nr_{k}}\psi ({\textbf {w}}^{\circ }){\frac {d{\textbf {w}}^{\circ }}{{\textbf {w}}^{\circ }}}}
where
ψ
(
w
∘
)
=
R
e
s
(
F
(
w
∘
,
w
k
)
w
k
;
w
k
=
g
(
w
∘
)
)
{\displaystyle \psi ({\textbf {w}}^{\circ })=Res\left({\frac {F({\textbf {w}}^{\circ },w_{k})}{w_{k}}};w_{k}=g({\textbf {w}}^{\circ })\right)}
.
Because
w
{\displaystyle {\textbf {w}}}
is a minimal point, the domain of convergence of the integral is greater than
ρ
{\displaystyle \rho }
away from
w
{\displaystyle {\textbf {w}}}
.
a
n
r
−
χ
=
1
(
2
π
i
)
d
−
1
∫
T
∘
∖
N
(
w
∘
)
−
n
r
∘
g
(
w
∘
)
−
n
r
k
ψ
(
w
∘
)
d
w
∘
w
∘
<
|
w
−
n
r
|
{\displaystyle a_{n{\textbf {r}}}-\chi ={\frac {1}{(2\pi i)^{d-1}}}\int _{T^{\circ }\setminus {\mathcal {N}}}({{\textbf {w}}^{\circ }})^{-n{\textbf {r}}^{\circ }}g({\textbf {w}}^{\circ })^{-nr_{k}}\psi ({\textbf {w}}^{\circ }){\frac {d{\textbf {w}}^{\circ }}{{\textbf {w}}^{\circ }}}<|{\textbf {w}}^{-n{\textbf {r}}}|}
By changing variables
z
j
=
w
j
e
i
θ
j
(
1
≤
j
≤
d
,
j
≠
k
)
{\displaystyle z_{j}={\textbf {w}}_{j}e^{i\theta _{j}}\quad (1\leq j\leq d,j\neq k)}
. Let
N
′
{\displaystyle {\mathcal {N'}}}
be the image of
N
{\displaystyle {\mathcal {N}}}
under this change of variables. This is a neighbourhood of the origin in
R
d
−
1
{\displaystyle \mathbb {R} ^{d-1}}
.
g
{\displaystyle g}
and
ψ
{\displaystyle \psi }
can be re-written after this change of variables[18]
ϕ
(
θ
)
=
l
o
g
g
(
w
∘
)
e
i
θ
g
(
w
∘
)
+
i
r
k
r
∘
.
θ
{\displaystyle \phi (\theta )=log{\frac {g({\textbf {w}}^{\circ })e^{i\theta }}{g({\textbf {w}}^{\circ })}}+{\frac {i}{r_{k}}}{\textbf {r}}^{\circ }.\theta }
Therefore,
χ
{\displaystyle \chi }
can be written[19]
χ
=
1
(
2
π
)
d
−
1
w
−
n
r
∫
N
′
e
−
n
r
k
ϕ
(
θ
)
ψ
(
w
∘
e
i
θ
)
d
θ
{\displaystyle \chi ={\frac {1}{(2\pi )^{d-1}}}{\textbf {w}}^{-n{\textbf {r}}}\int _{\mathcal {N'}}e^{-nr_{k}\phi (\theta )}\psi ({\textbf {w}}^{\circ }e^{i\theta })d\theta }
By Fourier-Laplace integrals [20]
∫
N
′
e
−
n
r
k
ϕ
(
θ
)
ψ
(
w
∘
e
i
θ
)
d
θ
∼
(
2
π
)
(
d
−
1
)
/
2
det
H
ψ
(
w
∘
e
i
0
)
(
n
r
k
)
(
1
−
d
)
/
2
=
(
2
π
)
(
d
−
1
)
/
2
det
H
G
(
w
)
w
k
∂
H
∂
w
k
(
w
)
(
n
r
k
)
(
1
−
d
)
/
2
{\displaystyle \int _{\mathcal {N'}}e^{-nr_{k}\phi (\theta )}\psi ({\textbf {w}}^{\circ }e^{i\theta })d\theta \sim {\frac {(2\pi )^{(d-1)/2}}{\sqrt {\det {\mathcal {H}}}}}\psi ({\textbf {w}}^{\circ }e^{i0})(nr_{k})^{(1-d)/2}={\frac {(2\pi )^{(d-1)/2}}{\sqrt {\det {\mathcal {H}}}}}{\frac {G({\textbf {w}})}{w_{k}{\frac {\partial H}{\partial w_{k}}}({\textbf {w}})}}(nr_{k})^{(1-d)/2}}
Therefore
χ
∼
w
−
n
r
(
2
π
)
(
1
−
d
)
/
2
det
H
G
(
w
)
w
k
∂
H
∂
w
k
(
w
)
(
n
r
k
)
(
1
−
d
)
/
2
{\displaystyle \chi \sim {\textbf {w}}^{-n{\textbf {r}}}{\frac {(2\pi )^{(1-d)/2}}{\sqrt {\det {\mathcal {H}}}}}{\frac {G({\textbf {w}})}{w_{k}{\frac {\partial H}{\partial w_{k}}}({\textbf {w}})}}(nr_{k})^{(1-d)/2}}
Theorem
If
f
(
z
∘
,
y
)
{\displaystyle f({\textbf {z}}^{\circ },y)}
is a holomorphic function at
w
∈
C
d
{\displaystyle {\textbf {w}}\in \mathbb {C} ^{d}}
and
∂
f
∂
y
(
w
)
≠
0
{\displaystyle {\frac {\partial f}{\partial y}}({\textbf {w}})\neq 0}
then for
z
∘
{\displaystyle {\textbf {z}}^{\circ }}
in a neighbourhood of
w
∘
{\displaystyle {\textbf {w}}^{\circ }}
there is a unique holomorphic function
g
(
z
∘
)
{\displaystyle g({\textbf {z}}^{\circ })}
such that
f
(
z
∘
,
y
)
=
0
⟺
y
=
g
(
z
∘
)
{\displaystyle f({\textbf {z}}^{\circ },y)=0\iff y=g({\textbf {z}}^{\circ })}
.[21]
Proof
The solution to
f
(
z
∘
,
g
)
=
0
{\displaystyle f({\textbf {z}}^{\circ },g)=0}
exists(?) and can be differentiated with respect to
z
¯
j
(
j
=
2
,
⋯
,
d
)
{\displaystyle {\bar {z}}_{j}(j=2,\cdots ,d)}
in the form
∂
f
∂
z
1
∂
g
∂
z
¯
j
+
∂
f
∂
z
¯
1
∂
g
∂
z
j
¯
+
∂
f
∂
z
¯
j
=
0
{\displaystyle {\frac {\partial f}{\partial z_{1}}}{\frac {\partial g}{\partial {\bar {z}}_{j}}}+{\frac {\partial f}{\partial {\bar {z}}_{1}}}{\bar {\frac {\partial g}{\partial z_{j}}}}+{\frac {\partial f}{\partial {\bar {z}}_{j}}}=0}
Because
f
{\displaystyle f}
is holomorphic the Cauchy-Riemann conditions apply such that
∂
f
∂
z
¯
1
=
∂
f
∂
z
¯
j
=
0
{\displaystyle {\frac {\partial f}{\partial {\bar {z}}_{1}}}={\frac {\partial f}{\partial {\bar {z}}_{j}}}=0}
leaving us with
∂
f
∂
z
1
∂
g
∂
z
¯
j
=
0
{\displaystyle {\frac {\partial f}{\partial z_{1}}}{\frac {\partial g}{\partial {\bar {z}}_{j}}}=0}
But, by the hypothesis
∂
f
∂
y
(
w
)
≠
0
{\displaystyle {\frac {\partial f}{\partial y}}({\textbf {w}})\neq 0}
, therefore
∂
g
∂
z
¯
j
=
0.
{\displaystyle {\frac {\partial g}{\partial {\bar {z}}_{j}}}=0.}
[22]
Bear in mind that the neighbourhood
N
{\displaystyle {\mathcal {N}}}
is a necessary condition otherwise we might have multiple poles...
Projection in this context is straight-forward. Simply remove the
k
t
h
{\displaystyle kth}
coordinate from
w
{\displaystyle {\textbf {w}}}
to give
w
∘
=
(
w
1
,
⋯
,
w
k
−
1
,
w
k
+
1
,
⋯
,
w
d
)
∈
C
∗
d
−
1
{\displaystyle {\textbf {w}}^{\circ }=(w_{1},\cdots ,w_{k-1},w_{k+1},\cdots ,w_{d})\in \mathbb {C} _{*}^{d-1}}
. Similarly with
r
{\displaystyle {\textbf {r}}}
.
An integral
∫
f
(
x
,
y
)
d
x
d
y
{\displaystyle \int f(x,y)\,dx\,dy}
can be re-written
∫
(
∫
f
(
x
,
y
)
d
x
)
d
y
{\displaystyle \int \left(\int f(x,y)\,dx\right)dy}
, where in the inner integral
y
{\displaystyle y}
is kept constant.
We restrict all the coordinates to an arc containing
w
i
{\displaystyle w_{i}}
for each
w
i
∈
w
∘
{\displaystyle w_{i}\in {\textbf {w}}^{\circ }}
, i.e. all coordinates except the
k
t
h
{\displaystyle kth}
.
If
f
(
z
)
{\displaystyle f(z)}
is an analytic function on and inside the contour
C
{\displaystyle C}
except at a singularity
z
0
{\displaystyle z_{0}}
, then[23]
∫
C
f
(
z
)
d
z
=
(
2
π
i
)
R
e
s
(
f
(
z
)
;
z
=
z
0
)
{\displaystyle \int _{C}f(z)dz=(2\pi i)Res(f(z);z=z_{0})}
Theorem
If
A
{\displaystyle A}
and
ϕ
{\displaystyle \phi }
are complex-valued analytic functions on a compact neighbourhood
N
{\displaystyle {\mathcal {N}}}
of the origin in
R
d
{\displaystyle \mathbb {R} ^{d}}
, the real part of
ϕ
{\displaystyle \phi }
is nonnegative on
N
{\displaystyle {\mathcal {N}}}
and vanishes only at the origin and the Hessian matrix
H
{\displaystyle {\mathcal {H}}}
of
ϕ
{\displaystyle \phi }
at the origin is nonsingular, then[24]
∫
N
A
(
z
)
e
−
λ
ϕ
(
z
)
d
z
∼
A
(
0
)
(
2
π
)
d
/
2
det
H
λ
−
d
/
2
{\displaystyle \int _{\mathcal {N}}A(z)e^{-\lambda \phi (z)}dz\sim A(0){\frac {(2\pi )^{d/2}}{\sqrt {\det {\mathcal {H}}}}}\lambda ^{-d/2}}
Proof
By the complex Morse Lemma , there exists a change of variables
ψ
−
1
{\displaystyle \psi ^{-1}}
to move
N
{\displaystyle {\mathcal {N}}}
to a neighbourhood of the origin in
C
d
{\displaystyle \mathbb {C} ^{d}}
[25]
∫
N
A
(
z
)
e
−
λ
ϕ
(
z
)
d
z
=
∫
ψ
−
1
N
A
(
ψ
(
z
)
)
e
−
λ
S
(
z
)
(
det
d
ψ
(
z
)
)
d
z
{\displaystyle \int _{\mathcal {N}}A(z)e^{-\lambda \phi (z)}dz=\int _{\psi ^{-1}{\mathcal {N}}}A(\psi (z))e^{-\lambda S(z)}(\det d\psi (z))dz}
where
S
(
z
)
=
∑
i
=
1
d
z
i
2
{\displaystyle S(z)=\sum _{i=1}^{d}z_{i}^{2}}
.
Let
C
=
ψ
−
1
N
{\displaystyle C=\psi ^{-1}{\mathcal {N}}}
, a polydisk centred at the origin. Using the functions
I
d
(
z
)
=
z
{\displaystyle Id(z)=z}
and
π
(
z
)
=
R
e
{
z
}
{\displaystyle \pi (z)=Re\{z\}}
we construct a prism operator
P
{\displaystyle P}
which has the property[26]
∂
P
(
C
)
=
C
−
R
e
{
C
}
+
P
(
∂
C
)
{\displaystyle \partial P(C)=C-Re\{C\}+P(\partial C)}
Now, we can apply Stokes' formula
∫
∂
P
(
C
)
ω
=
0
{\displaystyle \int _{\partial P(C)}\omega =0}
which implies
∫
C
ω
=
∫
R
e
{
C
}
ω
−
∫
P
(
∂
C
)
ω
{\displaystyle \int _{C}\omega =\int _{Re\{C\}}\omega -\int _{P(\partial C)}\omega }
.
Therefore
∫
C
A
(
ψ
(
z
)
)
e
−
λ
S
(
z
)
(
det
d
ψ
(
z
)
)
d
z
=
∫
R
e
{
C
}
A
(
ψ
(
z
)
)
e
−
λ
S
(
z
)
(
det
d
ψ
(
z
)
)
d
z
+
O
(
e
−
ϵ
λ
)
{\displaystyle \int _{C}A(\psi (z))e^{-\lambda S(z)}(\det d\psi (z))dz=\int _{Re\{C\}}A(\psi (z))e^{-\lambda S(z)}(\det d\psi (z))dz+O(e^{-\epsilon \lambda })}
As proved in the complex Morse Lemma
det
d
ψ
(
0
)
=
2
d
/
2
det
H
{\displaystyle \det d\psi (0)={\frac {2^{d/2}}{\sqrt {\det {\mathcal {H}}}}}}
By applying the Gaussian integral (like in the univariate saddle-point method ) multiple times[27]
∫
π
(
C
)
e
−
λ
S
(
z
)
d
z
=
∏
j
=
1
d
∫
−
a
a
e
−
λ
S
(
z
)
d
z
∼
∏
j
=
1
d
∫
−
∞
∞
e
−
λ
S
(
z
)
d
z
=
∏
j
=
1
d
π
λ
=
π
d
/
2
λ
−
d
/
2
{\displaystyle \int _{\pi (C)}e^{-\lambda S(z)}dz=\prod _{j=1}^{d}\int _{-a}^{a}e^{-\lambda S(z)}dz\sim \prod _{j=1}^{d}\int _{-\infty }^{\infty }e^{-\lambda S(z)}dz=\prod _{j=1}^{d}{\sqrt {\frac {\pi }{\lambda }}}=\pi ^{d/2}\lambda ^{-d/2}}
Lemma
If
ϕ
(
x
)
{\displaystyle \phi (x)}
has vanishing gradient and nonsingular Hessian matrix
H
{\displaystyle {\mathcal {H}}}
at the origin then there exists a change of variables
x
=
ψ
(
y
)
{\displaystyle x=\psi (y)}
around
x
=
y
=
0
{\displaystyle x=y=0}
such that
ϕ
(
ψ
(
y
)
)
=
S
(
y
)
=
∑
j
=
1
d
y
j
2
{\displaystyle \phi (\psi (y))=S(y)=\sum _{j=1}^{d}y_{j}^{2}}
and
(
det
d
ψ
(
0
)
)
2
=
2
d
det
H
{\displaystyle (\det d\psi (0))^{2}={\frac {2^{d}}{\det {\mathcal {H}}}}}
.
Proof
Between two functions
f
,
g
:
X
→
Y
{\displaystyle f,g:X\to Y}
, a homotopy is a map
F
:
X
×
I
→
Y
{\displaystyle F:X\times I\to Y}
where
F
(
z
,
0
)
=
f
(
z
)
{\displaystyle F(z,0)=f(z)}
and
F
(
z
,
1
)
=
g
(
z
)
{\displaystyle F(z,1)=g(z)}
.
Explain chain...
From a homotopy and a chain
C
{\displaystyle C}
, we can define the prism operator [28]
P
(
C
)
=
∑
i
(
−
1
)
i
F
(
C
×
I
d
)
|
[
v
0
,
⋯
,
v
i
,
w
i
,
⋯
,
w
n
]
{\displaystyle P(C)=\sum _{i}(-1)^{i}F(C\times Id)|[v_{0},\cdots ,v_{i},w_{i},\cdots ,w_{n}]}
The prism operator satisfies the relation:[29]
∂
P
(
C
)
=
g
#
(
C
)
−
f
#
(
C
)
−
P
(
∂
C
)
{\displaystyle \partial P(C)=g_{\#}(C)-f_{\#}(C)-P(\partial C)}
where
f
#
{\displaystyle f_{\#}}
and
g
#
{\displaystyle g_{\#}}
map the chains in
X
{\displaystyle X}
to the chains in
Y
{\displaystyle Y}
.
If
M
{\displaystyle M}
is a complex manifold of dimension
n
{\displaystyle n}
,
ω
{\displaystyle \omega }
a holomorphic form of degree
n
{\displaystyle n}
and
σ
{\displaystyle \sigma }
an
(
n
+
1
)
{\displaystyle (n+1)}
-dimensional chain[30]
∫
∂
σ
ω
=
0
{\displaystyle \int _{\partial \sigma }\omega =0}
.
↑ Pemantle and Wilson 2013, pp. 215.
↑ Pemantle and Wilson 2013, pp. 163.
↑ Melczer 2021, pp. 206.
↑ Pemantle and Wilson 2013, pp. 164.
↑ Pemantle and Wilson 2013, pp. 341.
↑ Melczer 2021, pp. 213.
↑ Pemantle and Wilson 2013, pp. 169.
↑ Pemantle and Wilson 2013, pp. 8.
↑ Melczer 2021, pp. 201.
↑ Pemantle and Wilson 2013, pp. 164.
↑ Pemantle and Wilson 2013, pp. 165.
↑ Melczer 2021, pp. 206.
↑ Pemantle and Wilson 2013, pp. 164-165.
↑ Melczer 2021, pp. 206.
↑ Pemantle and Wilson 2013, pp. 164-165.
↑ Pemantle and Wilson 2013, pp. 165.
↑ Pemantle and Wilson 2013, pp. 166.
↑ Pemantle and Wilson 2013, pp. 167.
↑ Pemantle and Wilson 2013, pp. 167.
↑ Pemantle and Wilson 2013, pp. 89-90.
↑ Melczer 2021, pp. 97.
↑ Shabat 1992, pp. 41.
↑ Titchmarsh 1939, pp. 102.
↑ Pemantle, Wilson and Melczer 2024, pp. 131.
↑ Pemantle, Wilson and Melczer 2024, pp. 139.
↑ Hatcher 2001, pp. 112.
↑ Pemantle, Wilson and Melczer 2024, pp. 135.
↑ Hatcher 2001, pp. 112.
↑ Hatcher 2001, pp. 112.
↑ Shabat 1992, pp. 83.
Hatcher, Allen (2001). Algebraic Topology (PDF) . Cambridge University Press.
Melczer, Stephen (2021). An Invitation to Analytic Combinatorics: From One to Several Variables (PDF) . Springer Texts & Monographs in Symbolic Computation.
Mishna, Marni (2020). Analytic Combinatorics: A Multidimensional Approach . Taylor & Francis Group, LLC.
Pemantle, Robin; Wilson, Mark C. (2013). Analytic Combinatorics in Several Variables (PDF) (1st ed.). Cambridge University Press.
Pemantle, Robin; Wilson, Mark C.; Melczer, Stephen (2024). Analytic Combinatorics in Several Variables (PDF) (2nd ed.). Cambridge University Press.
Shabat, B. V. (1992). Introduction to Complex Analysis. Part II: Functions of Several Variables . American Mathematical Society, Providence, Rhode Island.
Titchmarsh, E. C. (1939). The Theory of Functions (2nd ed.). Oxford University Press.