User:Dom walden/Multivariate Analytic Combinatorics/Singular Varieties and Exponential Bounds

Introduction

As seen in Cauchy-Hadamard in the previous chapter, we want to find the ${\textbf {z}}$ in the domain/radii of convergence which minimises our estimate in the direction ${\textbf {r}}$.

We first present a way of thinking about domains of convergence using singular varieties.

We then present a general way of minimising ${\textbf {z}}$ using exponential bounds.

Singular varieties

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

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

[Draw singular variety for 1 - x - y?]

The domain of convergence of $F({\textbf {z}})$ is the complement of the singular variety.

${\mathcal {D}}=\mathbb {C} ^{d}\setminus {\mathcal {V}}$

${\mathcal {D}}$ may have multiple unconnected components. Each one is for a different Laurent series expansion.

In theory, we identify the component of the Laurent series we are interested in and find the point on the boundary of this component which minimises our estimate in the direction of r...

In practice, this can be hard because the component is not necessarily convex...

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

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

Again, this may leave us with multiple unconnected components, each one for a different Laurent series expansion. Denote the component we are interested in as $B$

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

Exponential bounds

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

• 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.