User:Dom walden/Multivariate Analytic Combinatorics/Smooth Point via Surgery

Critical points

We attach a few conditions to the critical point ${\textbf {w}}$ and the function $H$ of this chapter. Later chapters relax or modify these conditions.

Squarefree

$H$ can be factored into $H_{1}H_{2}\cdots H_{n}$ where no factor has a power and no factor is repeated.

Minimality

The critical point ${\textbf {w}}$ is one of

• Strictly minimal: the only critical point on its polytorus, i.e.

${\mathcal {V}}\cap T({\textbf {w}})=\{{\textbf {w}}\}$

• Finitely minimal: one of a finite number of critical point on its polytorus, i.e.

$1<\#({\mathcal {V}}\cap T({\textbf {w}}))<\infty$

• Torally minimal: one of an infinite number of critical point on its polytorus, i.e.

$\#({\mathcal {V}}\cap T({\textbf {w}}))=\infty$

and the the torality hypothesis is satisfied...

Smoothly varying

At least one of the partial derivatives ${\frac {\partial H}{\partial w_{k}}}({\textbf {w}})\neq 0$.

Formally, it means the gradient map of $H$ at ${\textbf {w}}$ is not equal to the zero vector.

$\nabla H({\textbf {w}})\neq {\textbf {0}}$

where:

$\nabla f(p)={\begin{bmatrix}{\frac {\partial f}{\partial x_{1}}}(p)\\\vdots \\{\frac {\partial f}{\partial x_{n}}}(p)\end{bmatrix}}$

Quadratically nondegenerate

In the proof, we make use of the Hessian matrix ${\mathcal {H}}$

${\mathcal {H}}=\left[{\frac {\partial ^{2}\phi ({\textbf {w}})}{\partial z_{i}\partial z_{j}}}\right]$

where $\phi$ is a function we will see later.

Quadratic nondegeneracy means the Hessian matrix is nonsingular, a fact we use in the proof.

Theorems

One quadratically nondegenerate smooth point

Let $F({\textbf {z}})={\frac {G({\textbf {z}})}{H({\textbf {z}})}}$ where $H(z)$ is squarefree and has a strictly minimal, smoothly varying and quadratically nondegenerate critical point ${\textbf {w}}=(w_{1},\cdots ,w_{d})\in \mathbb {C} _{*}^{d}$ in the direction ${\textbf {r}}=(r_{1},\cdots ,r_{d})\in \mathbb {N} _{*}^{d}$. Then,

$a_{n{\textbf {r}}}\sim {\textbf {w}}^{-n{\textbf {r}}}{\frac {(2\pi )^{(1-d)/2}}{\sqrt {\det {\mathcal {H}}}}}{\frac {G({\textbf {w}})}{w_{k}{\frac {\partial H}{\partial w_{k}}}({\textbf {w}})}}(nr_{k})^{(1-d)/2}$

where ${\mathcal {H}}$ is the Hessian matrix of $\phi (\theta )$ at ${\textbf {w}}$.

Proof

By the multivariate Cauchy formula

$a_{n{\textbf {r}}}={\frac {1}{(2\pi i)^{d}}}\int _{T}F({\textbf {z}}){\frac {d{\textbf {z}}}{{\textbf {z}}^{n{\textbf {r}}+1}}}$

To make use of the implicit function theorem, we choose one coordinate from the critical point for which ${\frac {\partial H}{\partial w_{k}}}({\textbf {w}})\neq 0$, possible because it is smooth. Call this coordinate $w_{k}$. Call the projection of the critical point ${\textbf {w}}$ into $d-1$ coordinates by ${\textbf {w}}^{\circ }$ and the projection of the direction ${\textbf {r}}$ into $d-1$ coordinates by ${\textbf {r}}^{\circ }$.

Rewrite the above Cauchy formula as an iterated integral, defining $T^{\circ }$ as the torus through ${\textbf {w}}^{\circ }$

$a_{n{\textbf {r}}}={\frac {1}{(2\pi i)^{d}}}\int _{T^{\circ }}({{\textbf {w}}^{\circ }})^{-n{\textbf {r}}^{\circ }}\left(\int _{|w_{k}|=\rho -\delta _{s}}w_{k}^{-nr_{k}}F({\textbf {w}}^{\circ },w_{k}){\frac {dw_{k}}{w_{k}}}\right){\frac {d{\textbf {w}}^{\circ }}{{\textbf {w}}^{\circ }}}$

By the implicit function theorem, there exists a neighbourhood ${\mathcal {N}}$ of ${\textbf {w}}^{\circ }$ such that $H({\textbf {w}}^{\circ },z)=0\iff z=g({\textbf {w}}^{\circ })$

$I={\frac {1}{(2\pi i)^{d}}}\int _{\mathcal {N}}({{\textbf {w}}^{\circ }})^{-n{\textbf {r}}^{\circ }}\left(\int _{|w_{k}|=\rho -\delta _{s}}w_{k}^{-nr_{k}}F({\textbf {w}}^{\circ },w_{k}){\frac {dw_{k}}{w_{k}}}\right){\frac {d{\textbf {w}}^{\circ }}{{\textbf {w}}^{\circ }}}$

and

$I'={\frac {1}{(2\pi i)^{d}}}\int _{\mathcal {N}}({\textbf {w}}^{\circ })^{-n{\textbf {r}}^{\circ }}\left(\int _{|w_{k}|=\rho +\delta _{s}}w_{k}^{-nr_{k}}F({\textbf {w}}^{\circ },w_{k}){\frac {dw_{k}}{w_{k}}}\right){\frac {d{\textbf {w}}^{\circ }}{{\textbf {w}}^{\circ }}}$

As a result of the implicit function theorem, for any fixed ${\textbf {w}}^{\circ }$ there is a unique pole a $z=g({\textbf {w}}^{\circ })$ inside the annulus.

The difference between the two inner integrals of $I$ and $I'$ is that the latter has this pole inside and therefore by integration with residues

$\int _{|w_{k}|=\rho +\delta _{s}}w_{k}^{-nr_{k}}F({\textbf {w}}^{\circ },w_{k}){\frac {dw_{k}}{w_{k}}}-\int _{|w_{k}|=\rho -\delta _{s}}w_{k}^{-nr_{k}}F({\textbf {w}}^{\circ },w_{k}){\frac {dw_{k}}{w_{k}}}=(2\pi i)Res\left(g({\textbf {w}}^{\circ })^{-nr_{k}}{\frac {F({\textbf {w}}^{\circ },w_{k})}{w_{k}}};w_{k}=g({\textbf {w}}^{\circ })\right)$

therefore

$\chi =I-I'={\frac {1}{(2\pi i)^{d-1}}}\int _{\mathcal {N}}({{\textbf {w}}^{\circ }})^{-n{\textbf {r}}^{\circ }}g({\textbf {w}}^{\circ })^{-nr_{k}}\psi ({\textbf {w}}^{\circ }){\frac {d{\textbf {w}}^{\circ }}{{\textbf {w}}^{\circ }}}$

where $\psi ({\textbf {w}}^{\circ })=Res\left({\frac {F({\textbf {w}}^{\circ },w_{k})}{w_{k}}};w_{k}=g({\textbf {w}}^{\circ })\right)$.

Because ${\textbf {w}}$ is a minimal point, the domain of convergence of the integral is greater than $\rho$ away from ${\textbf {w}}$.

$a_{n{\textbf {r}}}-\chi ={\frac {1}{(2\pi i)^{d-1}}}\int _{T^{\circ }\setminus {\mathcal {N}}}({{\textbf {w}}^{\circ }})^{-n{\textbf {r}}^{\circ }}g({\textbf {w}}^{\circ })^{-nr_{k}}\psi ({\textbf {w}}^{\circ }){\frac {d{\textbf {w}}^{\circ }}{{\textbf {w}}^{\circ }}}<|{\textbf {w}}^{-n{\textbf {r}}}|$

By changing variables $z_{j}={\textbf {w}}_{j}e^{i\theta _{j}}\quad (1\leq j\leq d,j\neq k)$. Let ${\mathcal {N'}}$ be the image of ${\mathcal {N}}$ under this change of variables. This is a neighbourhood of the origin in $\mathbb {R} ^{d-1}$. $g$ and $\psi$ can be re-written after this change of variables

$\phi (\theta )=log{\frac {g({\textbf {w}}^{\circ })e^{i\theta }}{g({\textbf {w}}^{\circ })}}+{\frac {i}{r_{k}}}{\textbf {r}}^{\circ }.\theta$

Therefore, $\chi$ can be written

$\chi ={\frac {1}{(2\pi )^{d-1}}}{\textbf {w}}^{-n{\textbf {r}}}\int _{\mathcal {N'}}e^{-nr_{k}\phi (\theta )}\psi ({\textbf {w}}^{\circ }e^{i\theta })d\theta$

By Fourier-Laplace integrals

$\int _{\mathcal {N'}}e^{-nr_{k}\phi (\theta )}\psi ({\textbf {w}}^{\circ }e^{i\theta })d\theta \sim {\frac {(2\pi )^{(d-1)/2}}{\sqrt {\det {\mathcal {H}}}}}\psi ({\textbf {w}}^{\circ }e^{i0})(nr_{k})^{(1-d)/2}={\frac {(2\pi )^{(d-1)/2}}{\sqrt {\det {\mathcal {H}}}}}{\frac {G({\textbf {w}})}{w_{k}{\frac {\partial H}{\partial w_{k}}}({\textbf {w}})}}(nr_{k})^{(1-d)/2}$

Therefore

$\chi \sim {\textbf {w}}^{-n{\textbf {r}}}{\frac {(2\pi )^{(1-d)/2}}{\sqrt {\det {\mathcal {H}}}}}{\frac {G({\textbf {w}})}{w_{k}{\frac {\partial H}{\partial w_{k}}}({\textbf {w}})}}(nr_{k})^{(1-d)/2}$

Definitions and lemmas

Implicit function theorem

Theorem

If $f({\textbf {z}}^{\circ },y)$ is a holomorphic function at ${\textbf {w}}\in \mathbb {C} ^{d}$ and ${\frac {\partial f}{\partial y}}({\textbf {w}})\neq 0$ then for ${\textbf {z}}^{\circ }$ in a neighbourhood of ${\textbf {w}}^{\circ }$ there is a unique holomorphic function $g({\textbf {z}}^{\circ })$ such that $f({\textbf {z}}^{\circ },y)=0\iff y=g({\textbf {z}}^{\circ })$.

Proof

The solution to $f({\textbf {z}}^{\circ },g)=0$ exists(?) and can be differentiated with respect to ${\bar {z}}_{j}(j=2,\cdots ,d)$ in the form

${\frac {\partial f}{\partial z_{1}}}{\frac {\partial g}{\partial {\bar {z}}_{j}}}+{\frac {\partial f}{\partial {\bar {z}}_{1}}}{\bar {\frac {\partial g}{\partial z_{j}}}}+{\frac {\partial f}{\partial {\bar {z}}_{j}}}=0$

Because $f$ is holomorphic the Cauchy-Riemann conditions apply such that ${\frac {\partial f}{\partial {\bar {z}}_{1}}}={\frac {\partial f}{\partial {\bar {z}}_{j}}}=0$ leaving us with

${\frac {\partial f}{\partial z_{1}}}{\frac {\partial g}{\partial {\bar {z}}_{j}}}=0$

But, by the hypothesis ${\frac {\partial f}{\partial y}}({\textbf {w}})\neq 0$, therefore ${\frac {\partial g}{\partial {\bar {z}}_{j}}}=0.$

Bear in mind that the neighbourhood ${\mathcal {N}}$ is a necessary condition otherwise we might have multiple poles...

Projection

Projection in this context is straight-forward. Simply remove the $kth$ coordinate from ${\textbf {w}}$ to give ${\textbf {w}}^{\circ }=(w_{1},\cdots ,w_{k-1},w_{k+1},\cdots ,w_{d})\in \mathbb {C} _{*}^{d-1}$. Similarly with ${\textbf {r}}$.

Iterated integral

An integral $\int f(x,y)\,dx\,dy$ can be re-written $\int \left(\int f(x,y)\,dx\right)dy$, where in the inner integral $y$ is kept constant.

Neighbourhood

We restrict all the coordinates to an arc containing $w_{i}$ for each $w_{i}\in {\textbf {w}}^{\circ }$, i.e. all coordinates except the $kth$.

Integration with residues

If $f(z)$ is an analytic function on and inside the contour $C$ except at a singularity $z_{0}$, then

$\int _{C}f(z)dz=(2\pi i)Res(f(z);z=z_{0})$

Fourier-Laplace integrals

Theorem

If $A$ and $\phi$ are complex-valued analytic functions on a compact neighbourhood ${\mathcal {N}}$ of the origin in $\mathbb {R} ^{d}$, the real part of $\phi$ is nonnegative on ${\mathcal {N}}$ and vanishes only at the origin and the Hessian matrix ${\mathcal {H}}$ of $\phi$ at the origin is nonsingular, then

$\int _{\mathcal {N}}A(z)e^{-\lambda \phi (z)}dz\sim A(0){\frac {(2\pi )^{d/2}}{\sqrt {\det {\mathcal {H}}}}}\lambda ^{-d/2}$

Proof

By the complex Morse Lemma, there exists a change of variables $\psi ^{-1}$ to move ${\mathcal {N}}$ to a neighbourhood of the origin in $\mathbb {C} ^{d}$

$\int _{\mathcal {N}}A(z)e^{-\lambda \phi (z)}dz=\int _{\psi ^{-1}{\mathcal {N}}}A(\psi (z))e^{-\lambda S(z)}(\det d\psi (z))dz$

where $S(z)=\sum _{i=1}^{d}z_{i}^{2}$.

Let $C=\psi ^{-1}{\mathcal {N}}$, a polydisk centred at the origin. Using the functions $Id(z)=z$ and $\pi (z)=Re\{z\}$ we construct a prism operator $P$ which has the property

$\partial P(C)=C-Re\{C\}+P(\partial C)$

Now, we can apply Stokes' formula

$\int _{\partial P(C)}\omega =0$

which implies

$\int _{C}\omega =\int _{Re\{C\}}\omega -\int _{P(\partial C)}\omega$.

Therefore

$\int _{C}A(\psi (z))e^{-\lambda S(z)}(\det d\psi (z))dz=\int _{Re\{C\}}A(\psi (z))e^{-\lambda S(z)}(\det d\psi (z))dz+O(e^{-\epsilon \lambda })$

As proved in the complex Morse Lemma

$\det d\psi (0)={\frac {2^{d/2}}{\sqrt {\det {\mathcal {H}}}}}$

By applying the Gaussian integral (like in the univariate saddle-point method) multiple times

$\int _{\pi (C)}e^{-\lambda S(z)}dz=\prod _{j=1}^{d}\int _{-a}^{a}e^{-\lambda S(z)}dz\sim \prod _{j=1}^{d}\int _{-\infty }^{\infty }e^{-\lambda S(z)}dz=\prod _{j=1}^{d}{\sqrt {\frac {\pi }{\lambda }}}=\pi ^{d/2}\lambda ^{-d/2}$

Complex Morse Lemma

Lemma

If $\phi (x)$ has vanishing gradient and nonsingular Hessian matrix ${\mathcal {H}}$ at the origin then there exists a change of variables $x=\psi (y)$ around $x=y=0$ such that $\phi (\psi (y))=S(y)=\sum _{j=1}^{d}y_{j}^{2}$ and $(\det d\psi (0))^{2}={\frac {2^{d}}{\det {\mathcal {H}}}}$.

Proof

Prism operator

Between two functions $f,g:X\to Y$, a homotopy is a map $F:X\times I\to Y$ where $F(z,0)=f(z)$ and $F(z,1)=g(z)$.

Explain chain...

From a homotopy and a chain $C$, we can define the prism operator

$P(C)=\sum _{i}(-1)^{i}F(C\times Id)|[v_{0},\cdots ,v_{i},w_{i},\cdots ,w_{n}]$

The prism operator satisfies the relation:

$\partial P(C)=g_{\#}(C)-f_{\#}(C)-P(\partial C)$

where $f_{\#}$ and $g_{\#}$ map the chains in $X$ to the chains in $Y$.

Stokes' formula

If $M$ is a complex manifold of dimension $n$, $\omega$ a holomorphic form of degree $n$ and $\sigma$ an $(n+1)$-dimensional chain

$\int _{\partial \sigma }\omega =0$.

References

• 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.