Introduction
edit
Complex analysis in several variables
edit
Vectors in multidimensional space
edit
The spaces
C
d
{\displaystyle \mathbb {C} ^{d}}
and
R
d
{\displaystyle \mathbb {R} ^{d}}
are made up of ordered sets of
n
{\displaystyle n}
complex or real (resp.) numbers
z
=
(
z
1
,
⋯
,
z
d
)
(
z
i
∈
C
)
{\displaystyle {\textbf {z}}=(z_{1},\cdots ,z_{d})\quad (z_{i}\in \mathbb {C} )}
x
=
(
x
1
,
⋯
,
x
d
)
(
x
i
∈
R
)
{\displaystyle {\textbf {x}}=(x_{1},\cdots ,x_{d})\quad (x_{i}\in \mathbb {R} )}
We write
z
∈
C
d
{\displaystyle {\textbf {z}}\in \mathbb {C} ^{d}}
or
x
∈
R
d
{\displaystyle {\textbf {x}}\in \mathbb {R} ^{d}}
.
Polydisk and polytorus
edit
For the vectors
a
∈
C
d
{\displaystyle {\textbf {a}}\in \mathbb {C} ^{d}}
and
r
∈
R
>
0
d
{\displaystyle {\textbf {r}}\in \mathbb {R} _{>0}^{d}}
the open polydisk centred at
a
{\displaystyle {\textbf {a}}}
of radius
r
{\displaystyle {\textbf {r}}}
[1]
D
a
(
r
)
=
{
z
∈
C
d
:
|
z
1
−
a
1
|
<
r
1
,
⋯
,
|
z
d
−
a
d
|
<
r
d
}
{\displaystyle D_{\textbf {a}}({\textbf {r}})=\{{\textbf {z}}\in \mathbb {C} ^{d}:|z_{1}-a_{1}|<r_{1},\cdots ,|z_{d}-a_{d}|<r_{d}\}}
and the polytorus
T
a
(
r
)
=
{
z
∈
C
d
:
|
z
1
−
a
1
|
=
r
1
,
⋯
,
|
z
d
−
a
d
|
=
r
d
}
{\displaystyle T_{\textbf {a}}({\textbf {r}})=\{{\textbf {z}}\in \mathbb {C} ^{d}:|z_{1}-a_{1}|=r_{1},\cdots ,|z_{d}-a_{d}|=r_{d}\}}
Multivariate Cauchy coefficient formula
edit
This is the multivariate version of the Cauchy coefficient formula .
By the multiple Cauchy integral[2]
f
(
z
)
=
1
(
2
π
i
)
n
∫
T
f
(
ζ
)
d
ζ
1
⋯
d
ζ
n
(
ζ
1
−
z
)
⋯
(
ζ
n
−
z
)
=
1
(
2
π
i
)
n
∫
T
f
(
ζ
)
d
ζ
(
ζ
−
z
)
{\displaystyle f(z)={\frac {1}{(2\pi i)^{n}}}\int _{T}{\frac {f(\zeta )d\zeta _{1}\cdots d\zeta _{n}}{(\zeta _{1}-z)\cdots (\zeta _{n}-z)}}={\frac {1}{(2\pi i)^{n}}}\int _{T}{\frac {f(\zeta )d\zeta }{(\zeta -z)}}}
and the fact that[3]
1
ζ
−
z
=
∑
|
k
|
=
0
∞
(
z
−
a
)
k
(
ζ
−
a
)
k
+
1
{\displaystyle {\frac {1}{\zeta -z}}=\sum _{|k|=0}^{\infty }{\frac {(z-a)^{k}}{(\zeta -a)^{k+1}}}}
we can re-write
f
(
z
)
{\displaystyle f(z)}
f
(
z
)
=
∑
|
k
|
=
0
∞
1
(
2
π
i
)
n
∫
T
f
(
ζ
)
d
ζ
(
ζ
−
a
)
k
+
1
(
z
−
a
)
k
=
∑
|
k
|
=
0
∞
c
k
(
z
−
a
)
k
{\displaystyle f(z)=\sum _{|k|=0}^{\infty }{\frac {1}{(2\pi i)^{n}}}\int _{T}{\frac {f(\zeta )d\zeta }{(\zeta -a)^{k+1}}}(z-a)^{k}=\sum _{|k|=0}^{\infty }c_{k}(z-a)^{k}}
where:
c
k
=
1
(
2
π
i
)
n
∫
T
f
(
ζ
)
d
ζ
(
ζ
−
a
)
k
+
1
{\displaystyle c_{k}={\frac {1}{(2\pi i)^{n}}}\int _{T}{\frac {f(\zeta )d\zeta }{(\zeta -a)^{k+1}}}}
Domain of convergence
edit
The domain of convergence of a power series is the set of points
z
∈
C
d
{\displaystyle {\textbf {z}}\in \mathbb {C} ^{d}}
such that the power series converges absolutely for some neighbourhood of
z
{\displaystyle {\textbf {z}}}
.[4]
The associated [5] or conjugate radii of convergence [6] are the vectors
r
∈
R
>
0
d
{\displaystyle {\textbf {r}}\in \mathbb {R} _{>0}^{d}}
such that the power series converges in the domain
{
z
∈
C
d
:
|
z
1
−
a
1
|
<
r
1
,
⋯
,
|
z
d
−
a
d
|
<
r
d
}
{\displaystyle \{{\textbf {z}}\in \mathbb {C} ^{d}:|z_{1}-a_{1}|<r_{1},\cdots ,|z_{d}-a_{d}|<r_{d}\}}
and diverges in the domain
{
z
∈
C
d
:
|
z
1
−
a
1
|
>
r
1
,
⋯
,
|
z
d
−
a
d
|
>
r
d
}
{\displaystyle \{{\textbf {z}}\in \mathbb {C} ^{d}:|z_{1}-a_{1}|>r_{1},\cdots ,|z_{d}-a_{d}|>r_{d}\}}
. Note that
r
{\displaystyle {\textbf {r}}}
is not necessarily unique and there may be infinite such
r
{\displaystyle {\textbf {r}}}
.
In our example,
1
1
−
x
−
y
{\displaystyle {\frac {1}{1-x-y}}}
, the denominator is zero for
0
≤
x
≤
1
{\displaystyle 0\leq x\leq 1}
and
y
=
1
−
x
{\displaystyle y=1-x}
.
Topology
edit
Algebraic topology
edit
Differential topology
edit
Multivariate generating functions
edit
Multivariate formal power series
edit
For a generating function in
d
{\displaystyle d}
variables.[7] [8]
z
=
(
z
1
,
⋯
,
z
d
)
∈
C
d
{\displaystyle {\textbf {z}}=(z_{1},\cdots ,z_{d})\in \mathbb {C} ^{d}}
,
n
=
(
n
1
,
⋯
,
n
d
)
∈
N
d
{\displaystyle {\textbf {n}}=(n_{1},\cdots ,n_{d})\in \mathbb {N} ^{d}}
and
z
n
=
z
1
n
1
⋯
z
d
n
d
{\displaystyle {\textbf {z}}^{\textbf {n}}=z_{1}^{n_{1}}\cdots z_{d}^{n_{d}}}
The multivariate formal power series
F
(
z
)
=
∑
n
∈
N
d
f
n
z
n
=
∑
(
n
1
,
⋯
,
n
d
)
∈
N
d
f
n
1
,
⋯
,
n
d
z
1
n
1
⋯
z
d
n
d
{\displaystyle F({\textbf {z}})=\sum _{{\textbf {n}}\in \mathbb {N} ^{d}}f_{\textbf {n}}{\textbf {z}}^{\textbf {n}}=\sum _{(n_{1},\cdots ,n_{d})\in \mathbb {N} ^{d}}f_{n_{1},\cdots ,n_{d}}z_{1}^{n_{1}}\cdots z_{d}^{n_{d}}}
For example[9]
1
1
−
x
−
y
=
∑
(
n
,
m
)
∈
N
2
(
n
+
m
m
)
x
m
y
n
=
∑
n
≥
0
∑
m
≥
0
(
n
+
m
m
)
x
m
y
n
{\displaystyle {\frac {1}{1-x-y}}=\sum _{(n,m)\in \mathbb {N} ^{2}}{\binom {n+m}{m}}x^{m}y^{n}=\sum _{n\geq 0}\sum _{m\geq 0}{\binom {n+m}{m}}x^{m}y^{n}}
Diagonals
edit
The central diagonal of a power series[10]
Δ
F
(
z
)
=
∑
n
≥
0
f
n
,
⋯
,
n
z
n
{\displaystyle \Delta F({\textbf {z}})=\sum _{n\geq 0}f_{n,\cdots ,n}z^{n}}
For our example power series,
1
1
−
x
−
y
{\displaystyle {\frac {1}{1-x-y}}}
, we can represent it in two dimensions like below. The central diagonal is highlighted in green.
x
0
{\displaystyle x^{0}}
x
1
{\displaystyle x^{1}}
x
2
{\displaystyle x^{2}}
x
3
{\displaystyle x^{3}}
x
4
{\displaystyle x^{4}}
y
0
{\displaystyle y^{0}}
1
x
0
y
0
{\displaystyle 1x^{0}y^{0}}
1
x
1
y
0
{\displaystyle 1x^{1}y^{0}}
1
x
2
y
0
{\displaystyle 1x^{2}y^{0}}
1
x
3
y
0
{\displaystyle 1x^{3}y^{0}}
1
x
4
y
0
{\displaystyle 1x^{4}y^{0}}
y
1
{\displaystyle y^{1}}
1
x
0
y
1
{\displaystyle 1x^{0}y^{1}}
2
x
1
y
1
{\displaystyle 2x^{1}y^{1}}
3
x
2
y
1
{\displaystyle 3x^{2}y^{1}}
4
x
3
y
1
{\displaystyle 4x^{3}y^{1}}
5
x
4
y
1
{\displaystyle 5x^{4}y^{1}}
y
2
{\displaystyle y^{2}}
1
x
0
y
2
{\displaystyle 1x^{0}y^{2}}
3
x
1
y
2
{\displaystyle 3x^{1}y^{2}}
6
x
2
y
2
{\displaystyle 6x^{2}y^{2}}
10
x
3
y
2
{\displaystyle 10x^{3}y^{2}}
15
x
4
y
2
{\displaystyle 15x^{4}y^{2}}
y
3
{\displaystyle y^{3}}
1
x
0
y
3
{\displaystyle 1x^{0}y^{3}}
4
x
1
y
3
{\displaystyle 4x^{1}y^{3}}
10
x
2
y
3
{\displaystyle 10x^{2}y^{3}}
20
x
3
y
3
{\displaystyle 20x^{3}y^{3}}
35
x
4
y
3
{\displaystyle 35x^{4}y^{3}}
y
4
{\displaystyle y^{4}}
1
x
0
y
4
{\displaystyle 1x^{0}y^{4}}
5
x
1
y
4
{\displaystyle 5x^{1}y^{4}}
15
x
2
y
4
{\displaystyle 15x^{2}y^{4}}
35
x
3
y
4
{\displaystyle 35x^{3}y^{4}}
70
x
4
y
4
{\displaystyle 70x^{4}y^{4}}
We can generalise this to a diagonal along a ray r ,[11] where
r
=
(
r
1
,
⋯
,
r
d
)
∈
N
d
{\displaystyle {\textbf {r}}=(r_{1},\cdots ,r_{d})\in \mathbb {N} ^{d}}
Δ
r
F
(
z
)
=
∑
n
≥
0
f
n
r
1
,
⋯
,
n
r
d
z
n
{\displaystyle \Delta ^{\textbf {r}}F({\textbf {z}})=\sum _{n\geq 0}f_{nr_{1},\cdots ,nr_{d}}z^{n}}
For example
Δ
(
2
,
1
)
1
1
−
x
−
y
=
∑
n
≥
0
(
2
n
+
n
n
)
z
n
{\displaystyle \Delta ^{(2,1)}{\frac {1}{1-x-y}}=\sum _{n\geq 0}{\binom {2n+n}{n}}z^{n}}
which is represented below in green.
x
0
{\displaystyle x^{0}}
x
1
{\displaystyle x^{1}}
x
2
{\displaystyle x^{2}}
x
3
{\displaystyle x^{3}}
x
4
{\displaystyle x^{4}}
y
0
{\displaystyle y^{0}}
1
x
0
y
0
{\displaystyle 1x^{0}y^{0}}
1
x
1
y
0
{\displaystyle 1x^{1}y^{0}}
1
x
2
y
0
{\displaystyle 1x^{2}y^{0}}
1
x
3
y
0
{\displaystyle 1x^{3}y^{0}}
1
x
4
y
0
{\displaystyle 1x^{4}y^{0}}
y
1
{\displaystyle y^{1}}
1
x
0
y
1
{\displaystyle 1x^{0}y^{1}}
2
x
1
y
1
{\displaystyle 2x^{1}y^{1}}
3
x
2
y
1
{\displaystyle 3x^{2}y^{1}}
4
x
3
y
1
{\displaystyle 4x^{3}y^{1}}
5
x
4
y
1
{\displaystyle 5x^{4}y^{1}}
y
2
{\displaystyle y^{2}}
1
x
0
y
2
{\displaystyle 1x^{0}y^{2}}
3
x
1
y
2
{\displaystyle 3x^{1}y^{2}}
6
x
2
y
2
{\displaystyle 6x^{2}y^{2}}
10
x
3
y
2
{\displaystyle 10x^{3}y^{2}}
15
x
4
y
2
{\displaystyle 15x^{4}y^{2}}
y
3
{\displaystyle y^{3}}
1
x
0
y
3
{\displaystyle 1x^{0}y^{3}}
4
x
1
y
3
{\displaystyle 4x^{1}y^{3}}
10
x
2
y
3
{\displaystyle 10x^{2}y^{3}}
20
x
3
y
3
{\displaystyle 20x^{3}y^{3}}
35
x
4
y
3
{\displaystyle 35x^{4}y^{3}}
y
4
{\displaystyle y^{4}}
1
x
0
y
4
{\displaystyle 1x^{0}y^{4}}
5
x
1
y
4
{\displaystyle 5x^{1}y^{4}}
15
x
2
y
4
{\displaystyle 15x^{2}y^{4}}
35
x
3
y
4
{\displaystyle 35x^{3}y^{4}}
70
x
4
y
4
{\displaystyle 70x^{4}y^{4}}
Cauchy-Hadamard in several complex variables
edit
Let
α
{\displaystyle \alpha }
be an n -dimensional vector of natural numbers (
α
=
(
α
1
,
⋯
,
α
n
)
∈
N
n
{\displaystyle \alpha =(\alpha _{1},\cdots ,\alpha _{n})\in \mathbb {N} ^{n}}
) with
|
|
α
|
|
=
α
1
+
⋯
+
α
n
{\displaystyle ||\alpha ||=\alpha _{1}+\cdots +\alpha _{n}}
, then
f
(
x
)
{\displaystyle f(x)}
converges with radius of convergence
ρ
=
(
ρ
1
,
⋯
,
ρ
n
)
∈
R
n
{\displaystyle \rho =(\rho _{1},\cdots ,\rho _{n})\in \mathbb {R} ^{n}}
with
ρ
α
=
ρ
1
α
1
⋯
ρ
n
α
n
{\displaystyle \rho ^{\alpha }=\rho _{1}^{\alpha _{1}}\cdots \rho _{n}^{\alpha _{n}}}
if and only if
lim sup
|
|
α
|
|
→
∞
|
c
α
|
ρ
α
|
|
α
|
|
=
1
{\displaystyle \limsup _{||\alpha ||\to \infty }{\sqrt[{||\alpha ||}]{|c_{\alpha }|\rho ^{\alpha }}}=1}
to the multidimensional power series
∑
α
≥
0
c
α
(
z
−
a
)
α
:=
∑
α
1
≥
0
,
…
,
α
n
≥
0
c
α
1
,
…
,
α
n
(
z
1
−
a
1
)
α
1
⋯
(
z
n
−
a
n
)
α
n
{\displaystyle \sum _{\alpha \geq 0}c_{\alpha }(z-a)^{\alpha }:=\sum _{\alpha _{1}\geq 0,\ldots ,\alpha _{n}\geq 0}c_{\alpha _{1},\ldots ,\alpha _{n}}(z_{1}-a_{1})^{\alpha _{1}}\cdots (z_{n}-a_{n})^{\alpha _{n}}}
Set
z
=
a
+
t
ρ
{\displaystyle z=a+t\rho }
(
z
i
=
a
i
+
t
ρ
i
)
{\displaystyle (z_{i}=a_{i}+t\rho _{i})}
, then[12]
∑
α
≥
0
c
α
(
z
−
a
)
α
=
∑
α
≥
0
c
α
ρ
α
t
|
|
α
|
|
=
∑
μ
≥
0
(
∑
|
|
α
|
|
=
μ
|
c
α
|
ρ
α
)
t
μ
{\displaystyle \sum _{\alpha \geq 0}c_{\alpha }(z-a)^{\alpha }=\sum _{\alpha \geq 0}c_{\alpha }\rho ^{\alpha }t^{||\alpha ||}=\sum _{\mu \geq 0}\left(\sum _{||\alpha ||=\mu }|c_{\alpha }|\rho ^{\alpha }\right)t^{\mu }}
This is a power series in one variable
t
{\displaystyle t}
which converges for
|
t
|
<
1
{\displaystyle |t|<1}
and diverges for
|
t
|
>
1
{\displaystyle |t|>1}
. Therefore, by the Cauchy-Hadamard theorem for one variable
lim sup
μ
→
∞
∑
|
|
α
|
|
=
μ
|
c
α
|
ρ
α
μ
=
1
{\displaystyle \limsup _{\mu \to \infty }{\sqrt[{\mu }]{\sum _{||\alpha ||=\mu }|c_{\alpha }|\rho ^{\alpha }}}=1}
Setting
|
c
m
|
ρ
m
=
max
|
|
α
|
|
=
μ
|
c
α
|
ρ
α
{\displaystyle |c_{m}|\rho ^{m}=\max _{||\alpha ||=\mu }|c_{\alpha }|\rho ^{\alpha }}
gives us an estimate
|
c
m
|
ρ
m
≤
∑
|
|
α
|
|
=
μ
|
c
α
|
ρ
α
≤
(
μ
+
1
)
n
|
c
m
|
ρ
m
{\displaystyle |c_{m}|\rho ^{m}\leq \sum _{||\alpha ||=\mu }|c_{\alpha }|\rho ^{\alpha }\leq (\mu +1)^{n}|c_{m}|\rho ^{m}}
Because
(
μ
+
1
)
n
μ
→
1
{\displaystyle {\sqrt[{\mu }]{(\mu +1)^{n}}}\to 1}
as
μ
→
∞
{\displaystyle \mu \to \infty }
|
c
m
|
ρ
m
μ
≤
∑
|
|
α
|
|
=
μ
|
c
α
|
ρ
α
μ
≤
|
c
m
|
ρ
m
μ
⟹
∑
|
|
α
|
|
=
μ
|
c
α
|
ρ
α
μ
=
|
c
m
|
ρ
m
μ
(
μ
→
∞
)
{\displaystyle {\sqrt[{\mu }]{|c_{m}|\rho ^{m}}}\leq {\sqrt[{\mu }]{\sum _{||\alpha ||=\mu }|c_{\alpha }|\rho ^{\alpha }}}\leq {\sqrt[{\mu }]{|c_{m}|\rho ^{m}}}\implies {\sqrt[{\mu }]{\sum _{||\alpha ||=\mu }|c_{\alpha }|\rho ^{\alpha }}}={\sqrt[{\mu }]{|c_{m}|\rho ^{m}}}\qquad (\mu \to \infty )}
Therefore
lim sup
|
|
α
|
|
→
∞
|
c
α
|
ρ
α
|
α
|
=
lim sup
μ
→
∞
|
c
m
|
ρ
m
μ
=
1
{\displaystyle \limsup _{||\alpha ||\to \infty }{\sqrt[{|\alpha |}]{|c_{\alpha }|\rho ^{\alpha }}}=\limsup _{\mu \to \infty }{\sqrt[{\mu }]{|c_{m}|\rho ^{m}}}=1}
For the central diagonal of our example,
Δ
1
1
−
x
−
y
=
∑
n
≥
0
f
n
,
n
x
n
y
n
{\displaystyle \Delta {\frac {1}{1-x-y}}=\sum _{n\geq 0}f_{n,n}x^{n}y^{n}}
:
lim sup
n
→
∞
|
f
n
,
n
|
x
n
y
n
n
=
1
⟹
lim sup
n
→
∞
|
f
n
,
n
|
=
1
(
x
y
)
n
{\displaystyle \limsup _{n\to \infty }{\sqrt[{n}]{|f_{n,n}|x^{n}y^{n}}}=1\implies \limsup _{n\to \infty }|f_{n,n}|={\frac {1}{(xy)^{n}}}}
x
n
y
n
{\displaystyle x^{n}y^{n}}
is at its largest when
x
=
y
=
1
2
{\displaystyle x=y={\frac {1}{2}}}
so that
lim sup
n
→
∞
|
f
n
,
n
|
=
4
n
{\displaystyle \limsup _{n\to \infty }|f_{n,n}|=4^{n}}
.
We know by Stirling's approximation that this is a good estimate.
But what about a diagonal along an arbitrary ray, like the above example
Δ
(
2
,
1
)
1
1
−
x
−
y
{\displaystyle \Delta ^{(2,1)}{\frac {1}{1-x-y}}}
?
lim sup
|
n
r
|
→
∞
|
f
2
n
,
n
|
x
2
n
y
n
|
n
r
|
=
1
⟹
lim sup
|
n
r
|
→
∞
|
f
2
n
,
n
|
=
1
(
x
2
y
)
n
{\displaystyle \limsup _{|n{\textbf {r}}|\to \infty }{\sqrt[{|n{\textbf {r}}|}]{|f_{2n,n}|x^{2n}y^{n}}}=1\implies \limsup _{|n{\textbf {r}}|\to \infty }|f_{2n,n}|={\frac {1}{(x^{2}y)^{n}}}}
If we keep
x
=
y
=
1
2
{\displaystyle x=y={\frac {1}{2}}}
then
lim sup
|
n
r
|
→
∞
|
f
2
n
,
n
|
=
8
n
{\displaystyle \limsup _{|n{\textbf {r}}|\to \infty }|f_{2n,n}|=8^{n}}
This isn't a good estimate.
Better to use
x
=
2
3
,
y
=
1
3
{\displaystyle x={\frac {2}{3}},y={\frac {1}{3}}}
then
lim sup
|
n
r
|
→
∞
|
f
2
n
,
n
|
=
(
27
4
)
n
=
(
6.75
)
n
{\displaystyle \limsup _{|n{\textbf {r}}|\to \infty }|f_{2n,n}|=\left({\frac {27}{4}}\right)^{n}=(6.75)^{n}}
Exponential bounds
edit
We need a general way of finding the
z
{\displaystyle {\textbf {z}}}
which minimises our estimate in the direction
r
{\displaystyle {\textbf {r}}}
.
Assuming
F
(
z
)
=
G
(
z
)
H
(
z
)
{\displaystyle F({\textbf {z}})={\frac {G({\textbf {z}})}{H({\textbf {z}})}}}
The set of points
z
{\displaystyle z}
where
H
(
z
)
=
0
{\displaystyle H(z)=0}
is called the singular variety
V
{\displaystyle {\mathcal {V}}}
.
Define[13]
R
e
l
o
g
(
z
)
=
(
log
|
z
1
|
,
⋯
,
log
|
z
d
|
)
{\displaystyle Relog({\textbf {z}})=(\log |z_{1}|,\cdots ,\log |z_{d}|)}
a
m
o
e
b
a
(
H
)
=
{
R
e
l
o
g
(
z
)
:
H
(
z
)
=
0
}
{\displaystyle amoeba(H)=\{Relog({\textbf {z}}):H({\textbf {z}})=0\}}
[Example of amoeba for 1 - x - y]
The domain of convergence of our function can now be defined as the complement of this amoeba[14]
a
m
o
e
b
a
(
H
)
c
=
R
d
∖
a
m
o
e
b
a
(
H
)
{\displaystyle amoeba(H)^{c}=\mathbb {R} ^{d}\setminus amoeba(H)}
Note that this may leave us with multiple unconnected components. Each one is for a different Laurent series expansion. Denote the component we are interested in as
B
{\displaystyle B}
D
=
R
e
l
o
g
−
1
(
B
)
{\displaystyle {\mathcal {D}}=Relog^{-1}(B)}
Minimising
w
r
{\displaystyle {\textbf {w}}^{\textbf {r}}}
is difficult as it is a nonlinear function. It is easier to minimise a linear function[15] [16]
h
r
(
z
)
=
−
∑
j
=
1
d
r
j
log
|
z
j
|
{\displaystyle h_{\textbf {r}}({\textbf {z}})=-\sum _{j=1}^{d}r_{j}\log |z_{j}|}
We define
∇
log
H
(
w
)
=
(
w
1
∂
H
(
w
)
∂
w
1
,
⋯
,
w
d
∂
H
(
w
)
∂
w
d
)
{\displaystyle \nabla _{\log }H({\textbf {w}})=\left(w_{1}{\frac {\partial H({\textbf {w}})}{\partial w_{1}}},\cdots ,w_{d}{\frac {\partial H({\textbf {w}})}{\partial w_{d}}}\right)}
Lemma If
w
∈
V
∩
∂
D
{\displaystyle {\textbf {w}}\in {\mathcal {V}}\cap \partial {\mathcal {D}}}
and
∇
log
H
(
w
)
=
τ
r
{\displaystyle \nabla _{\log }H({\textbf {w}})=\tau {\textbf {r}}}
(
τ
≠
0
)
{\displaystyle (\tau \neq 0)}
then
w
{\displaystyle {\textbf {w}}}
is either a maximiser or minimiser for
h
r
(
z
)
{\displaystyle h_{\textbf {r}}({\textbf {z}})}
on
D
¯
{\displaystyle {\overline {\mathcal {D}}}}
.[17]
Proof
Therefore, to find the minimiser of
h
r
(
z
)
{\displaystyle h_{\textbf {r}}({\textbf {z}})}
we need to find the conditions under which
∇
log
H
(
w
)
{\displaystyle \nabla _{\log }H({\textbf {w}})}
is a scalar multiple of
r
{\displaystyle {\textbf {r}}}
.[18] This means they are not linearly independent and therefore the matrix
(
w
1
∂
H
∂
z
1
(
w
)
⋯
w
d
∂
H
∂
z
d
(
w
)
r
1
⋯
r
d
)
{\displaystyle {\begin{pmatrix}w_{1}{\frac {\partial H}{\partial z_{1}}}({\textbf {w}})&\cdots &w_{d}{\frac {\partial H}{\partial z_{d}}}({\textbf {w}})\\r_{1}&\cdots &r_{d}\end{pmatrix}}}
is rank deficient, or its 2 x 2 submatrices have zero determinants. This is equivalent to a system of equations referred to as the critical point equations [19] [20] [21]
H
(
w
)
=
0
r
j
w
1
∂
H
∂
z
1
(
w
)
−
r
1
w
j
∂
H
∂
z
j
(
w
)
=
0
(
2
≤
j
≤
d
)
.
{\displaystyle H({\textbf {w}})=0\quad r_{j}w_{1}{\frac {\partial H}{\partial z_{1}}}({\textbf {w}})-r_{1}w_{j}{\frac {\partial H}{\partial z_{j}}}({\textbf {w}})=0\quad (2\leq j\leq d).}
But, even after finding our
z
{\displaystyle {\textbf {z}}}
, the estimate is only a bound and it may not be tight.[22]
↑ Melczer 2021, pp. 94.
↑ Shabat 1992, pp. 18.
↑ Shabat 1992, pp. 19.
↑ Melczer 2021, pp. 100.
↑ Fuks 1963, pp. 46.
↑ Shabat 1992, pp. 32.
↑ Melczer 2021, pp. 93.
↑ Mishna 2020, pp. 56.
↑ Mishna 2020, pp. 142-145.
↑ Mishna 2020, pp. 56-57.
↑ Mishna 2020, pp. 57.
↑ Shabat 1992, pp. 32-33.
↑ Pemantle and Wilson 2013, pp. 120, 127.
↑ Melczer 2021, pp. 116.
↑ Mishna 2020, pp. 146.
↑ Melczer 2021, pp. 202.
↑ Melczer 2021, pp. 202.
↑ Melczer 2021, pp. 203.
↑ Melczer 2021, pp. 203.
↑ Pemantle and Wilson 2013, pp. 145.
↑ Mishna 2020, pp. 147.
↑ Pemantle, Wilson and Melczer 2024, pp. 177.
References
edit
Fuks, B. A. (1963). Theory of Analytic Functions of Several Complex Variables . American Mathematical Society, Providence, Rhode Island.
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.; 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.
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