resulting object is again a cube. In this manner, we can use cubes to cover the solutions to $ F=0$, and then refine the cover using successively smaller cubes.

Secondly, cubes facilitate explicit computations of $F_M$ and analytical estimates of $F_\infty$. While formally $F_M$ is an infinite dimensional map, computationally, we may consider $F_M$ to be a map $ {\mathbb{R}}^2 \times {\mathbb{C}}^M \to {\mathbb{C}}^M$. To calculate $F_M$, we simply truncate the second sum in at $ j = M-k$. As the $\pi'_M$ projection of a cube is given as a finite product of intervals, it is well suited for using interval arithmetic [@moore2009introduction] to bound the image of $F_M(X)$. On the other hand, bounding $F_\infty$ requires significantly more analysis. Below is a simple, yet ever recurring estimate in our calculations: $$\label{eq:SumIntegral}
     \sum_{k=M+1}^\infty \frac{1}{k^s} \leq \int_M^\infty \frac{1}{x^s} dx = \frac{1}{(s-1)M^{s-1}},$$ where we take $ s >1$. For example, if a cube $ X \subseteq {\mathbb{R}}^2 \times {\Omega}^s$ satisfies $ s >1$, then $ \| \pi_c x \|_{\ell^1} \leq   2 \sum_{k=1}^M |X|_k  + \frac{2 C_0 }{(s-1)M^{s-1}} $ for all $ x \in X$. This specific bound on the $\ell^1$ norm is later used in Algorithm \[alg:Prune\] to check whether Lemmas \[prop:zeroneighborhood2\] or 