User:Dom walden/Multivariate Analytic Combinatorics/Basics

Complex analysis in several variables

Vectors in multidimensional space

The spaces $\mathbb {C} ^{d}$ and $\mathbb {R} ^{d}$ are made up of ordered sets of $n$ complex or real (resp.) numbers

${\textbf {z}}=(z_{1},\cdots ,z_{d})\quad (z_{i}\in \mathbb {C} )$

${\textbf {x}}=(x_{1},\cdots ,x_{d})\quad (x_{i}\in \mathbb {R} )$

We write ${\textbf {z}}\in \mathbb {C} ^{d}$ or ${\textbf {x}}\in \mathbb {R} ^{d}$.

Polydisk and polytorus

For the vectors ${\textbf {a}}\in \mathbb {C} ^{d}$ and ${\textbf {r}}\in \mathbb {R} _{>0}^{d}$ the open polydisk centred at ${\textbf {a}}$ of radius ${\textbf {r}}$

$D_{\textbf {a}}({\textbf {r}})=\{{\textbf {z}}\in \mathbb {C} ^{d}:|z_{1}-a_{1}|<r_{1},\cdots ,|z_{d}-a_{d}|<r_{d}\}$

and the polytorus

$T_{\textbf {a}}({\textbf {r}})=\{{\textbf {z}}\in \mathbb {C} ^{d}:|z_{1}-a_{1}|=r_{1},\cdots ,|z_{d}-a_{d}|=r_{d}\}$

Multivariate Cauchy coefficient formula

This is the multivariate version of the Cauchy coefficient formula.

By the multiple Cauchy integral

$f(z)={\frac {1}{(2\pi i)^{n}}}\int _{T}{\frac {f(\zeta )d\zeta _{1}\cdots d\zeta _{n}}{(\zeta _{1}-z)\cdots (\zeta _{n}-z)}}={\frac {1}{(2\pi i)^{n}}}\int _{T}{\frac {f(\zeta )d\zeta }{(\zeta -z)}}$

and the fact that

${\frac {1}{\zeta -z}}=\sum _{|k|=0}^{\infty }{\frac {(z-a)^{k}}{(\zeta -a)^{k+1}}}$

we can re-write $f(z)$

$f(z)=\sum _{|k|=0}^{\infty }{\frac {1}{(2\pi i)^{n}}}\int _{T}{\frac {f(\zeta )d\zeta }{(\zeta -a)^{k+1}}}(z-a)^{k}=\sum _{|k|=0}^{\infty }c_{k}(z-a)^{k}$

where:

$c_{k}={\frac {1}{(2\pi i)^{n}}}\int _{T}{\frac {f(\zeta )d\zeta }{(\zeta -a)^{k+1}}}$

Domain of convergence

The domain of convergence of a power series is the set of points ${\textbf {z}}\in \mathbb {C} ^{d}$ such that the power series converges absolutely for some neighbourhood of ${\textbf {z}}$.

The associated or conjugate radii of convergence are the vectors ${\textbf {r}}\in \mathbb {R} _{>0}^{d}$ such that the power series converges in the domain $\{{\textbf {z}}\in \mathbb {C} ^{d}:|z_{1}-a_{1}|<r_{1},\cdots ,|z_{d}-a_{d}|<r_{d}\}$ and diverges in the domain $\{{\textbf {z}}\in \mathbb {C} ^{d}:|z_{1}-a_{1}|>r_{1},\cdots ,|z_{d}-a_{d}|>r_{d}\}$. Note that ${\textbf {r}}$ is not necessarily unique and there may be infinite such ${\textbf {r}}$.

In our example, ${\frac {1}{1-x-y}}$, the denominator is zero for $0\leq x\leq 1$ and $y=1-x$.

Multivariate generating functions

Multivariate formal power series

For a generating function in $d$ variables.

${\textbf {z}}=(z_{1},\cdots ,z_{d})\in \mathbb {C} ^{d}$, ${\textbf {n}}=(n_{1},\cdots ,n_{d})\in \mathbb {N} ^{d}$ and ${\textbf {z}}^{\textbf {n}}=z_{1}^{n_{1}}\cdots z_{d}^{n_{d}}$

The multivariate formal power series

$F({\textbf {z}})=\sum _{{\textbf {n}}\in \mathbb {N} ^{d}}f_{\textbf {n}}{\textbf {z}}^{\textbf {n}}=\sum _{(n_{1},\cdots ,n_{d})\in \mathbb {N} ^{d}}f_{n_{1},\cdots ,n_{d}}z_{1}^{n_{1}}\cdots z_{d}^{n_{d}}$

For example

${\frac {1}{1-x-y}}=\sum _{(n,m)\in \mathbb {N} ^{2}}{\binom {n+m}{m}}x^{m}y^{n}=\sum _{n\geq 0}\sum _{m\geq 0}{\binom {n+m}{m}}x^{m}y^{n}$

Diagonals

The central diagonal of a power series

$\Delta F({\textbf {z}})=\sum _{n\geq 0}f_{n,\cdots ,n}z^{n}$

For our example power series, ${\frac {1}{1-x-y}}$, we can represent it in two dimensions like below. The central diagonal is highlighted in green.

	$x^{0}$	$x^{1}$	$x^{2}$	$x^{3}$	$x^{4}$
$y^{0}$	$1x^{0}y^{0}$	$1x^{1}y^{0}$	$1x^{2}y^{0}$	$1x^{3}y^{0}$	$1x^{4}y^{0}$
$y^{1}$	$1x^{0}y^{1}$	$2x^{1}y^{1}$	$3x^{2}y^{1}$	$4x^{3}y^{1}$	$5x^{4}y^{1}$
$y^{2}$	$1x^{0}y^{2}$	$3x^{1}y^{2}$	$6x^{2}y^{2}$	$10x^{3}y^{2}$	$15x^{4}y^{2}$
$y^{3}$	$1x^{0}y^{3}$	$4x^{1}y^{3}$	$10x^{2}y^{3}$	$20x^{3}y^{3}$	$35x^{4}y^{3}$
$y^{4}$	$1x^{0}y^{4}$	$5x^{1}y^{4}$	$15x^{2}y^{4}$	$35x^{3}y^{4}$	$70x^{4}y^{4}$

We can generalise this to a diagonal along a ray r, where ${\textbf {r}}=(r_{1},\cdots ,r_{d})\in \mathbb {N} ^{d}$

$\Delta ^{\textbf {r}}F({\textbf {z}})=\sum _{n\geq 0}f_{nr_{1},\cdots ,nr_{d}}z^{n}$

For example

$\Delta ^{(2,1)}{\frac {1}{1-x-y}}=\sum _{n\geq 0}{\binom {2n+n}{n}}z^{n}$

which is represented below in green.

	$x^{0}$	$x^{1}$	$x^{2}$	$x^{3}$	$x^{4}$
$y^{0}$	$1x^{0}y^{0}$	$1x^{1}y^{0}$	$1x^{2}y^{0}$	$1x^{3}y^{0}$	$1x^{4}y^{0}$
$y^{1}$	$1x^{0}y^{1}$	$2x^{1}y^{1}$	$3x^{2}y^{1}$	$4x^{3}y^{1}$	$5x^{4}y^{1}$
$y^{2}$	$1x^{0}y^{2}$	$3x^{1}y^{2}$	$6x^{2}y^{2}$	$10x^{3}y^{2}$	$15x^{4}y^{2}$
$y^{3}$	$1x^{0}y^{3}$	$4x^{1}y^{3}$	$10x^{2}y^{3}$	$20x^{3}y^{3}$	$35x^{4}y^{3}$
$y^{4}$	$1x^{0}y^{4}$	$5x^{1}y^{4}$	$15x^{2}y^{4}$	$35x^{3}y^{4}$	$70x^{4}y^{4}$

Cauchy-Hadamard in several complex variables

Theorem

Let $\alpha$ be an n-dimensional vector of natural numbers ($\alpha =(\alpha _{1},\cdots ,\alpha _{n})\in \mathbb {N} ^{n}$) with $||\alpha ||=\alpha _{1}+\cdots +\alpha _{n}$, then $f(x)$ converges with radius of convergence $\rho =(\rho _{1},\cdots ,\rho _{n})\in \mathbb {R} ^{n}$ with $\rho ^{\alpha }=\rho _{1}^{\alpha _{1}}\cdots \rho _{n}^{\alpha _{n}}$ if and only if

$\limsup _{||\alpha ||\to \infty }{\sqrt[{||\alpha ||}]{|c_{\alpha }|\rho ^{\alpha }}}=1$

to the multidimensional power series

$\sum _{\alpha \geq 0}c_{\alpha }(z-a)^{\alpha }:=\sum _{\alpha _{1}\geq 0,\ldots ,\alpha _{n}\geq 0}c_{\alpha _{1},\ldots ,\alpha _{n}}(z_{1}-a_{1})^{\alpha _{1}}\cdots (z_{n}-a_{n})^{\alpha _{n}}$

Proof

Set $z=a+t\rho$ $(z_{i}=a_{i}+t\rho _{i})$, then

$\sum _{\alpha \geq 0}c_{\alpha }(z-a)^{\alpha }=\sum _{\alpha \geq 0}c_{\alpha }\rho ^{\alpha }t^{||\alpha ||}=\sum _{\mu \geq 0}\left(\sum _{||\alpha ||=\mu }|c_{\alpha }|\rho ^{\alpha }\right)t^{\mu }$

This is a power series in one variable $t$ which converges for $|t|<1$ and diverges for $|t|>1$. Therefore, by the Cauchy-Hadamard theorem for one variable

$\limsup _{\mu \to \infty }{\sqrt[{\mu }]{\sum _{||\alpha ||=\mu }|c_{\alpha }|\rho ^{\alpha }}}=1$

Setting $|c_{m}|\rho ^{m}=\max _{||\alpha ||=\mu }|c_{\alpha }|\rho ^{\alpha }$ gives us an estimate

$|c_{m}|\rho ^{m}\leq \sum _{||\alpha ||=\mu }|c_{\alpha }|\rho ^{\alpha }\leq (\mu +1)^{n}|c_{m}|\rho ^{m}$

Because ${\sqrt[{\mu }]{(\mu +1)^{n}}}\to 1$ as $\mu \to \infty$

${\sqrt[{\mu }]{|c_{m}|\rho ^{m}}}\leq {\sqrt[{\mu }]{\sum _{||\alpha ||=\mu }|c_{\alpha }|\rho ^{\alpha }}}\leq {\sqrt[{\mu }]{|c_{m}|\rho ^{m}}}\implies {\sqrt[{\mu }]{\sum _{||\alpha ||=\mu }|c_{\alpha }|\rho ^{\alpha }}}={\sqrt[{\mu }]{|c_{m}|\rho ^{m}}}\qquad (\mu \to \infty )$

Therefore

$\limsup _{||\alpha ||\to \infty }{\sqrt[{|\alpha |}]{|c_{\alpha }|\rho ^{\alpha }}}=\limsup _{\mu \to \infty }{\sqrt[{\mu }]{|c_{m}|\rho ^{m}}}=1$

Example

For the central diagonal of our example, $\Delta {\frac {1}{1-x-y}}=\sum _{n\geq 0}f_{n,n}x^{n}y^{n}$:

$\limsup _{n\to \infty }{\sqrt[{n}]{|f_{n,n}|x^{n}y^{n}}}=1\implies \limsup _{n\to \infty }|f_{n,n}|={\frac {1}{(xy)^{n}}}$

$x^{n}y^{n}$ is at its largest when $x=y={\frac {1}{2}}$ so that $\limsup _{n\to \infty }|f_{n,n}|=4^{n}$.

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 $\Delta ^{(2,1)}{\frac {1}{1-x-y}}$?

$\limsup _{|n{\textbf {r}}|\to \infty }{\sqrt[{|n{\textbf {r}}|}]{|f_{2n,n}|x^{2n}y^{n}}}=1\implies \limsup _{|n{\textbf {r}}|\to \infty }|f_{2n,n}|={\frac {1}{(x^{2}y)^{n}}}$

If we keep $x=y={\frac {1}{2}}$ then $\limsup _{|n{\textbf {r}}|\to \infty }|f_{2n,n}|=8^{n}$

This isn't a good estimate.

Better to use $x={\frac {2}{3}},y={\frac {1}{3}}$ then $\limsup _{|n{\textbf {r}}|\to \infty }|f_{2n,n}|=\left({\frac {27}{4}}\right)^{n}=(6.75)^{n}$

Exponential bounds

We need a general way of finding the ${\textbf {z}}$ which minimises our estimate in the direction ${\textbf {r}}$.

Assuming $F({\textbf {z}})={\frac {G({\textbf {z}})}{H({\textbf {z}})}}$

The set of points $z$ where $H(z)=0$ is called the singular variety ${\mathcal {V}}$.

Define

$Relog({\textbf {z}})=(\log |z_{1}|,\cdots ,\log |z_{d}|)$

$amoeba(H)=\{Relog({\textbf {z}}):H({\textbf {z}})=0\}$

[Example of amoeba for 1 - x - y]

The domain of convergence of our function can now be defined as the complement of this amoeba

$amoeba(H)^{c}=\mathbb {R} ^{d}\setminus amoeba(H)$

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 $B$

${\mathcal {D}}=Relog^{-1}(B)$

Minimising ${\textbf {w}}^{\textbf {r}}$ is difficult as it is a nonlinear function. It is easier to minimise a linear function

$h_{\textbf {r}}({\textbf {z}})=-\sum _{j=1}^{d}r_{j}\log |z_{j}|$

We define

$\nabla _{\log }H({\textbf {w}})=\left(w_{1}{\frac {\partial H({\textbf {w}})}{\partial w_{1}}},\cdots ,w_{d}{\frac {\partial H({\textbf {w}})}{\partial w_{d}}}\right)$

Lemma If ${\textbf {w}}\in {\mathcal {V}}\cap \partial {\mathcal {D}}$ and $\nabla _{\log }H({\textbf {w}})=\tau {\textbf {r}}$ $(\tau \neq 0)$ then ${\textbf {w}}$ is either a maximiser or minimiser for $h_{\textbf {r}}({\textbf {z}})$ on ${\overline {\mathcal {D}}}$.

Proof

Therefore, to find the minimiser of $h_{\textbf {r}}({\textbf {z}})$ we need to find the conditions under which $\nabla _{\log }H({\textbf {w}})$ is a scalar multiple of ${\textbf {r}}$. This means they are not linearly independent and therefore the matrix

${\begin{pmatrix}w_{1}{\frac {\partial H}{\partial z_{1}}}({\textbf {w}})&\cdots &w_{d}{\frac {\partial H}{\partial z_{d}}}({\textbf {w}})\\r_{1}&\cdots &r_{d}\end{pmatrix}}$

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

$H({\textbf {w}})=0\quad r_{j}w_{1}{\frac {\partial H}{\partial z_{1}}}({\textbf {w}})-r_{1}w_{j}{\frac {\partial H}{\partial z_{j}}}({\textbf {w}})=0\quad (2\leq j\leq d).$

But, even after finding our ${\textbf {z}}$, the estimate is only a bound and it may not be tight.

References

• 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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