User:Dom walden/Multivariate Analytic Combinatorics/Smooth Point via Residue Forms

Introduction
Here, we prove a theorem very similar to that of the previous chapter.

The proof is shorter but involves more advanced mathematics. It has the advantage that it does not assume the same minimality hypotheses.

Theorem
If $$F(z) = \frac{G(z)}{H(z)}$$ with a simple pole on $$\mathcal{V}$$. $$\mathcal{V}$$ is smooth above height $$h_\textbf{r} = c - \epsilon$$. If there is a subset $$W$$ of nondegenerate critical points on $$\mathcal{V}$$ at height $$c$$ such that $$C = \sum_{z \in W} C_*(z)$$ in $$H_d(\mathcal{M}, \mathcal{M}^{c-\epsilon})$$. Then, there is a compact neighbourhood $$\mathcal{N}$$ of $$\textbf{r}$$... Then


 * $$a_{n\textbf{r}} \sim \sum_{z \in W} \textbf{w}^{-n\textbf{r}} \frac{(2\pi)^{(1 - d)/2}}{\sqrt{\det \mathcal{H}}} \frac{G(\textbf{w})}{w_k H_k(\textbf{w})} (nr_k)^{(1 - d)/2}$$

Proof
Let $$B$$ and $$B'$$ be two components of $$amoeba(H)^c$$ where $$h_r$$ is not bounded from below on $$B'$$. Let $$T = T_e(x)$$ for some $$x \in B$$ and $$T' = T_e(x')$$ for some $$x' \in B'$$.

The intersection class $$INT(T, T')$$ is represented by the intersection of $$\mathcal{V}$$ and a homotopy between $$T$$ and $$T'$$ which intersects $$\mathcal{V}$$ transversely.

If we choose the homotopy such that its time $$t$$ cross-sections are tori that expand with $$t$$ and go through $$\textbf{w}$$, perturbed to intersect $$\mathcal{V}$$ transversely, then the class $$INT(T, T')$$ can be represented by a smooth (d - 1)-chain $$\gamma$$ on $$\mathcal{V}$$ on which the height reaches its maximum at $$\textbf{w}$$.

By the Cauchy coefficient formula and residue theorem:


 * $$\begin{align}

a_r &= \frac{1}{(2\pi i)^d} \int_T F(z) z^{-r-1} dz \\ &= \frac{1}{(2\pi i)^{d-1}} \int_\gamma Res(F(z) z^{-r-1} dz) + \frac{1}{(2\pi i)^{d-1}} \int_T' F(z) z^{-r-1} dz \\ &= \frac{1}{(2\pi i)^{d-1}} \int_\gamma Res(F(z) z^{-r-1} dz) \end{align}$$

As a result of ...


 * $$a_r = \frac{e^{-h_r(w)}}{(2\pi i)^{d-1}} \int_\gamma e^{-r_d \phi(z)} \frac{P(z)}{Q(z) \prod_{j=1}^d z_j} dz^\circ$$

By theorem 5.3 and the change of variables $$z_j = w_j e^{i\theta_j}$$ gives


 * $$a_{n\textbf{r}} \sim \sum_{z \in W} \textbf{w}^{-n\textbf{r}} \frac{(2\pi)^{(1 - d)/2}}{\sqrt{\det \mathcal{H}}} \frac{G(\textbf{w})}{w_k H_k(\textbf{w})} (nr_k)^{(1 - d)/2}$$