model – the point ${\mbox{\boldmath$\rho$}}= ({\ensuremath{\mathbf{1}}},1/|{\ensuremath{\mathcal{S}}}|)$ – and this uniformity aids greatly in computing the quantities required in the analysis of a general step set, in order to obtain Theorem \[thm:asm\].

We first verify our claim in the previous section that the variety is smooth (that is, at every point on ${\ensuremath{\mathcal{V}}}$ one of the partial derivatives $H_{z_k}$ or $H_t$ does not vanish). Indeed, any non-smooth point on ${\ensuremath{\mathcal{V}}}$ would have to satisfy both $$\begin{aligned}
1-t(z_1\cdots z_d)S({\ensuremath{\mathbf{z}}}) &= H = 0\\
\text{and} \quad -(z_1\cdots z_d)S({\ensuremath{\mathbf{z}}}) &= H_t = 0,\end{aligned}$$ which can never occur. Equivalently, this shows that at each point in ${\ensuremath{\mathcal{V}}}$ there exists a neighbourhood $N \subseteq \mathbb{C}^{d+1}$ such that ${\ensuremath{\mathcal{V}}}\cap N$ is a complex submanifold of $N$.

Critical points {#sec:crtpts}
---------------

The next step is to find the critical points. Determined through an appeal to stratified Morse theory, for a smooth variety the critical points are precisely those which satisfy the following *critical point equations*: $$H = 0, \quad tH_t = z_1H_{z_1},\quad tH_t = z_2H_{z_2},\quad \dots\quad   tH_t = z_dH_{z_d},$$ which we now solve. Given ${\ensuremath{\mathbf{z}}}\in \mathbb{C}^d$, define $${\ensuremath{\mathbf{z}}}_{\hat{k}} := (z_1,\dots,z_{k-1},z_{k+1},\dots,z_d) \in \mathbb{C}^{d-1}.$$ As each step in ${\ensuremath{\mathcal{S}}}$ has coordinates taking values in $\{-1,0,1\}$, we may collect the coefficients 