of the $k^\text{th}$ variable, and use the symmetries present to write $$S({\ensuremath{\mathbf{z}}})= ({\ensuremath{\overline{z}}}_k + z_k) S_{1}^{(k)}({\ensuremath{\mathbf{z}}}_{\hat{k}}) + S_{0}^{(k)}({\ensuremath{\mathbf{z}}}_{\hat{k}}), \label{eq:inv}$$ which uniquely defines the Laurent polynomials $S_1^{(k)}({\ensuremath{\mathbf{z}}}_{\hat{k}}) $ and $S_0^{(k)}({\ensuremath{\mathbf{z}}}_{\hat{k}}) $. With this notation the equation becomes $$t(z_1\cdots z_d)S({\ensuremath{\mathbf{z}}}) = t(z_1\cdots z_d)S({\ensuremath{\mathbf{z}}}) + t(z_1\cdots z_d)(z_kS_{z_k}({\ensuremath{\mathbf{z}}})),$$ which implies $$0 = t(z_1\cdots z_d)\cdot z_kS_{z_k}({\ensuremath{\mathbf{z}}})= t\left( z_k^2 -1\right)(z_1\cdots z_{k-1}z_{k+1}\cdots z_d)S_1^{(k)}({\ensuremath{\mathbf{z}}}_{\hat{k}}).  \label{eq:CPeqn}$$ Note that while $(z_1\cdots z_{k-1}z_{k+1}\cdots z_d)S_1^{(k)}({\ensuremath{\mathbf{z}}}_{\hat{k}})$ is a polynomial, $S_{1}^{(k)}({\ensuremath{\mathbf{z}}}_{\hat{k}})$ itself is a Laurent polynomial, so one must be careful when specializing variables to 0 in the expression. This calculation characterizes the critical points of ${\ensuremath{\mathcal{V}}}$.

\[prop:critpt\] The point $({\ensuremath{\mathbf{z}}},t) = (z_1,\dots,z_d,t) \in {\ensuremath{\mathcal{V}}}$ is a critical point of ${\ensuremath{\mathcal{V}}}$ if and only if for each $1\leq k\leq d$ either:

1.  $z_k= \pm1$ or,

2.  the polynomial $(y_1\cdots y_{k-1}y_{k+1}\cdots y_d)S_1^{(k)}({\ensuremath{\mathbf{y}}}_{\hat{k}})$ has a root at ${\ensuremath{\mathbf{z}}}$.

We have shown above that the critical point equations reduce to Equation . Furthermore, if $t$ were zero at a point on ${\ensuremath{\mathcal{V}}}$ then $0=H(z_1,\dots,z_n,0)=1$, a contradiction.

It is interesting to note that the polynomial $(y_1\cdots y_{k-1}y_{k+1}\cdots y_d)S_1^{(k)}({\ensuremath{\mathbf{y}}}_{\hat{k}})$ has combinatorial signifigance, as the subset of $S({\ensuremath{\mathbf{z}}})$ which encodes only the steps which move forwards in their $k^\text{th}$ coordinate.

Minimal points {#sec:minpts}
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Among the critical points, only those 