which are ‘closest’ to the origin will contribute to the asymptotics, up to an exponentially decaying error. This is analogous to the single variable case, where the singularities of minimum modulus are those which contribute to the dominant asymptotic term. To be precise, for any point $({\ensuremath{\mathbf{z}}},t)
\in\mathbb{C}^{d+1}$ we define the closed polydisk $$D({\ensuremath{\mathbf{z}}},t) := \{(\mathbf{w},t') \in \mathbb{C}^{d+1} : |t'|\leq|t|
\text{ and } |w_j|\leq|z_j| \text{ for } j=1,\dots,d\}.$$ The critical point $(\mathbf{z},t)$ is called *strictly minimal* if $D({\ensuremath{\mathbf{z}}},t)
\cap {\ensuremath{\mathcal{V}}}= \{({\ensuremath{\mathbf{z}}},t)\}$, and *finitely minimal* if the intersection contains only a finite number of points, all of which are on the boundary of $D({\ensuremath{\mathbf{z}}},t)$. Finally, we call a critical point *isolated* if there exists a neighbourhood of $\mathbb{C}^{d+1}$ where it is the only critical point. In our case, we need only be concerned with isolated finitely minimal points.

\[prop:minpt\] The point ${\mbox{\boldmath$\rho$}}= ({\ensuremath{\mathbf{1}}},1/|{\ensuremath{\mathcal{S}}}|)$ is a finitely minimal point of the variety ${\ensuremath{\mathcal{V}}}$. Furthermore, any point in $D({\mbox{\boldmath$\rho$}}) \cap {\ensuremath{\mathcal{V}}}$ is an isolated critical point.

The point ${\mbox{\boldmath$\rho$}}$ is critical as it lies on ${\ensuremath{\mathcal{V}}}$ and its first $d$ coordinates are all one. Suppose $({\ensuremath{\mathbf{w}}},t_{\ensuremath{\mathbf{w}}})$ lies in ${\ensuremath{\mathcal{V}}}\cap D({\mbox{\boldmath$\rho$}})$, where we note that any choice of ${\ensuremath{\mathbf{w}}}$ uniquely determines $t_{\ensuremath{\mathbf{w}}}$ on ${\ensuremath{\mathcal{V}}}$. 