${\ensuremath{\mathcal{S}}}$ contains a step which moves forward in the $k^\text{th}$ coordinate. This contradicts $({\ensuremath{\mathbf{w}}},t_{\ensuremath{\mathbf{w}}}) \in D({\mbox{\boldmath$\rho$}})$.

Asymptotics Results {#sec:results}
-------------------

To apply the formulas of [@PeWi02] we need to define a few quantities. To start, we note that on all of ${\ensuremath{\mathcal{V}}}$ we may parametrize the coordinate $t$ as $$t({\ensuremath{\mathbf{z}}}) = \frac{1}{z_1\cdots z_d S({\ensuremath{\mathbf{z}}})}.$$ For each point $({\ensuremath{\mathbf{w}}},t_{\ensuremath{\mathbf{w}}}) \in {\ensuremath{\mathcal{V}}}\cap D({\mbox{\boldmath$\rho$}})$, the analysis of [@PeWi02] shows that the asymptotics of the integral in question which determines asymptotics for a given model depends on the function $$\begin{aligned}
\tilde{f}^{({\ensuremath{\mathbf{w}}})}(\mbox{\boldmath$\theta$}) &= \log\left( \frac{t(w_1e^{i\theta_1},\dots,w_d e^{i\theta_d})}{t_{{\ensuremath{\mathbf{w}}}}}\right) + i \sum_{k=1}^d \theta_k \notag \\
&= \log\left( \frac{ S({\ensuremath{\mathbf{w}}})}{e^{i(\theta_1+\cdots+\theta_d)}S(w_1e^{i\theta_1},\dots,w_de^{i\theta_d})}\right) + i(\theta_1+\cdots+\theta_d) \notag \\
&= \log S({\ensuremath{\mathbf{w}}}) - \log S(w_1 e^{i\theta_1},\dots,w_d e^{i\theta_d}). \label{eq:ftil}\end{aligned}$$ Let $\mathcal{H}_{{\ensuremath{\mathbf{w}}}}$ denote the determinant of the Hessian of $\tilde{f}^{({\ensuremath{\mathbf{w}}})}(\mbox{\boldmath$\theta$})$ at $\mathbf{0}$: $$\mathcal{H}_{{\ensuremath{\mathbf{w}}}} := \det \tilde{f''}^{({\ensuremath{\mathbf{w}}})}(\mathbf{0}) = \begin{vmatrix} 
\tilde{f}^{({\ensuremath{\mathbf{w}}})}_{\theta_1 \theta_1}(\mathbf{0}) &  \tilde{f}^{({\ensuremath{\mathbf{w}}})}_{\theta_1 \theta_2}(\mathbf{0}) & \cdots & \tilde{f}^{({\ensuremath{\mathbf{w}}})}_{\theta_1 \theta_d}(\mathbf{0}) \\[+2mm]
\tilde{f}^{({\ensuremath{\mathbf{w}}})}_{\theta_2 \theta_1}(\mathbf{0}) &  \tilde{f}^{({\ensuremath{\mathbf{w}}})}_{\theta_2 \theta_2}(\mathbf{0}) & \cdots & \tilde{f}^{({\ensuremath{\mathbf{w}}})}_{\theta_2 \theta_d}(\mathbf{0}) \\[+2mm]
\vdots & \vdots & \ddots & \vdots \\[+2mm]
\tilde{f}^{({\ensuremath{\mathbf{w}}})}_{\theta_d \theta_1}(\mathbf{0}) &  \tilde{f}^{({\ensuremath{\mathbf{w}}})}_{\theta_d \theta_2}(\mathbf{0}) & \cdots & \tilde{f}^{({\ensuremath{\mathbf{w}}})}_{\theta_d \theta_d}(\mathbf{0}) 
 \end{vmatrix},$$ if $\mathcal{H}_{{\ensuremath{\mathbf{w}}}} \neq 0$ then we say $({\ensuremath{\mathbf{w}}},t_{\ensuremath{\mathbf{w}}})$ is *non-degenerate*. The main asymptotic result of smooth multivaritate analytic combinatorics, in this restricted context, is the following (the 