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

## Critical points

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

### Squarefree

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

### Minimality

The critical point ${\displaystyle {\textbf {w}}}$ is one of[2][3]

• Strictly minimal: the only critical point on its polytorus, i.e.
${\displaystyle {\mathcal {V}}\cap T({\textbf {w}})=\{{\textbf {w}}\}}$
• Finitely minimal: one of a finite number of critical point on its polytorus, i.e.
${\displaystyle 1<\#({\mathcal {V}}\cap T({\textbf {w}}))<\infty }$
• Torally minimal: one of an infinite number of critical point on its polytorus, i.e.
${\displaystyle \#({\mathcal {V}}\cap T({\textbf {w}}))=\infty }$
and the the torality hypothesis is satisfied...

### Smoothly varying

At least one of the partial derivatives ${\displaystyle {\frac {\partial H}{\partial w_{k}}}({\textbf {w}})\neq 0}$.[4]

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

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

where:

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

In the proof, we make use of the Hessian matrix ${\displaystyle {\mathcal {H}}}$[5]

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

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

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

where ${\displaystyle {\mathcal {H}}}$ is the Hessian matrix of ${\displaystyle \phi (\theta )}$ at ${\displaystyle {\textbf {w}}}$.[6][7]

## Proof

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

To make use of the implicit function theorem, we choose one coordinate from the critical point for which ${\displaystyle {\frac {\partial H}{\partial w_{k}}}({\textbf {w}})\neq 0}$, possible because it is smooth. Call this coordinate ${\displaystyle w_{k}}$. Call the projection of the critical point ${\displaystyle {\textbf {w}}}$ into ${\displaystyle d-1}$ coordinates by ${\displaystyle {\textbf {w}}^{\circ }}$ and the projection of the direction ${\displaystyle {\textbf {r}}}$ into ${\displaystyle d-1}$ coordinates by ${\displaystyle {\textbf {r}}^{\circ }}$.[10]

Rewrite the above Cauchy formula as an iterated integral, defining ${\displaystyle T^{\circ }}$ as the torus through ${\displaystyle {\textbf {w}}^{\circ }}$[11][12]

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

By the implicit function theorem, there exists a neighbourhood ${\displaystyle {\mathcal {N}}}$ of ${\displaystyle {\textbf {w}}^{\circ }}$ such that ${\displaystyle H({\textbf {w}}^{\circ },z)=0\iff z=g({\textbf {w}}^{\circ })}$[13][14]

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

and

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

As a result of the implicit function theorem, for any fixed ${\displaystyle {\textbf {w}}^{\circ }}$ there is a unique pole a ${\displaystyle z=g({\textbf {w}}^{\circ })}$ inside the annulus.[15]

The difference between the two inner integrals of ${\displaystyle I}$ and ${\displaystyle I'}$ is that the latter has this pole inside and therefore by integration with residues[16]

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

therefore[17]

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

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

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

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

By changing variables ${\displaystyle z_{j}={\textbf {w}}_{j}e^{i\theta _{j}}\quad (1\leq j\leq d,j\neq k)}$. Let ${\displaystyle {\mathcal {N'}}}$ be the image of ${\displaystyle {\mathcal {N}}}$ under this change of variables. This is a neighbourhood of the origin in ${\displaystyle \mathbb {R} ^{d-1}}$. ${\displaystyle g}$ and ${\displaystyle \psi }$ can be re-written after this change of variables[18]

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

Therefore, ${\displaystyle \chi }$ can be written[19]

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

Therefore

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

## Definitions and lemmas

### Implicit function theorem

Theorem

If ${\displaystyle f({\textbf {z}}^{\circ },y)}$ is a holomorphic function at ${\displaystyle {\textbf {w}}\in \mathbb {C} ^{d}}$ and ${\displaystyle {\frac {\partial f}{\partial y}}({\textbf {w}})\neq 0}$ then for ${\displaystyle {\textbf {z}}^{\circ }}$ in a neighbourhood of ${\displaystyle {\textbf {w}}^{\circ }}$ there is a unique holomorphic function ${\displaystyle g({\textbf {z}}^{\circ })}$ such that ${\displaystyle f({\textbf {z}}^{\circ },y)=0\iff y=g({\textbf {z}}^{\circ })}$.[21]

Proof

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

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

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

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

But, by the hypothesis ${\displaystyle {\frac {\partial f}{\partial y}}({\textbf {w}})\neq 0}$, therefore ${\displaystyle {\frac {\partial g}{\partial {\bar {z}}_{j}}}=0.}$[22]

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

### Projection

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

### Iterated integral

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

### Neighbourhood

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

### Integration with residues

If ${\displaystyle f(z)}$ is an analytic function on and inside the contour ${\displaystyle C}$ except at a singularity ${\displaystyle z_{0}}$, then[23]

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

### Fourier-Laplace integrals

Theorem

If ${\displaystyle A}$ and ${\displaystyle \phi }$ are complex-valued analytic functions on a compact neighbourhood ${\displaystyle {\mathcal {N}}}$ of the origin in ${\displaystyle \mathbb {R} ^{d}}$, the real part of ${\displaystyle \phi }$ is nonnegative on ${\displaystyle {\mathcal {N}}}$ and vanishes only at the origin and the Hessian matrix ${\displaystyle {\mathcal {H}}}$ of ${\displaystyle \phi }$ at the origin is nonsingular, then[24]

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

Proof

By the complex Morse Lemma, there exists a change of variables ${\displaystyle \psi ^{-1}}$ to move ${\displaystyle {\mathcal {N}}}$ to a neighbourhood of the origin in ${\displaystyle \mathbb {C} ^{d}}$[25]

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

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

Let ${\displaystyle C=\psi ^{-1}{\mathcal {N}}}$ be a polydisk centred at the origin. Using the functions ${\displaystyle Id(z)=z}$ and ${\displaystyle \pi (z)=Re\{z\}}$ we construct a prism operator ${\displaystyle P}$ which has the property[26]

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

Now, we can apply Stokes' formula

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

which implies

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

Therefore

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

As proved in the complex Morse Lemma

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

By applying the Gaussian integral (like in the univariate saddle-point method) multiple times[27]

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

### Complex Morse Lemma

Lemma

If ${\displaystyle \phi (x)}$ has vanishing gradient and nonsingular Hessian matrix ${\displaystyle {\mathcal {H}}}$ at the origin then there exists a change of variables ${\displaystyle x=\psi (y)}$ around ${\displaystyle x=y=0}$ such that ${\displaystyle \phi (\psi (y))=S(y)=\sum _{j=1}^{d}y_{j}^{2}}$ and ${\displaystyle (\det d\psi (0))^{2}={\frac {2^{d}}{\det {\mathcal {H}}}}}$.[28]

Proof

### Prism operator

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

Explain chain...

From a homotopy and a chain ${\displaystyle C}$, we can define the prism operator[29]

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

The prism operator satisfies the relation:[30]

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

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

### Stokes' formula

If ${\displaystyle M}$ is a complex manifold of dimension ${\displaystyle n}$, ${\displaystyle \omega }$ a holomorphic form of degree ${\displaystyle n}$ and ${\displaystyle \sigma }$ an ${\displaystyle (n+1)}$-dimensional chain[31]

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

## Notes

1. Pemantle and Wilson 2013, pp. 215.
2. Pemantle and Wilson 2013, pp. 163.
3. Melczer 2021, pp. 206.
4. Pemantle and Wilson 2013, pp. 164.
5. Pemantle and Wilson 2013, pp. 341.
6. Melczer 2021, pp. 213.
7. Pemantle and Wilson 2013, pp. 169.
8. Pemantle and Wilson 2013, pp. 8.
9. Melczer 2021, pp. 201.
10. Pemantle and Wilson 2013, pp. 164.
11. Pemantle and Wilson 2013, pp. 165.
12. Melczer 2021, pp. 206.
13. Pemantle and Wilson 2013, pp. 164-165.
14. Melczer 2021, pp. 206.
15. Pemantle and Wilson 2013, pp. 164-165.
16. Pemantle and Wilson 2013, pp. 165.
17. Pemantle and Wilson 2013, pp. 166.
18. Pemantle and Wilson 2013, pp. 167.
19. Pemantle and Wilson 2013, pp. 167.
20. Pemantle and Wilson 2013, pp. 89-90.
21. Melczer 2021, pp. 97.
22. Shabat 1992, pp. 41.
23. Titchmarsh 1939, pp. 102.
24. Pemantle, Wilson and Melczer 2024, pp. 131.
25. Pemantle, Wilson and Melczer 2024, pp. 139.
26. Hatcher 2001, pp. 112.
27. Pemantle, Wilson and Melczer 2024, pp. 135.
28. Pemantle, Wilson and Melczer 2024, pp. 137.
29. Hatcher 2001, pp. 112.
30. Hatcher 2001, pp. 112.
31. Shabat 1992, pp. 83.

## 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.