\times {\Omega}^s$ we consider in this paper are *cubes* which we define as follows:

\[def:cube\] For $M \in {\mathbb{N}}$, $s\geq 0$, $C_0>0$ define a cube $ X := X_M \times X_\infty \subseteq {\mathbb{R}}^2 \times \Omega^s$ to be of the following form: $$\begin{aligned}
    X_{M}       &:= [\underline{\alpha},\overline{\alpha}] \times [\underline{\omega},\overline{\omega}]   \times \prod_{k=1}^M [\underline{A}_k , \overline{A}_k] \times [\underline{B}_k , \overline{B}_k]  \label{eq:XMdef} \\   
    X_\infty    &:= \left\{ c_k  \in {\mathbb{C}}: |c_k| \leq C_0 /k^s  \right\}_{k=M+1}^\infty \label{eq:XIdef}. 
    \end{aligned}$$

To denote the union of a collection of cubes ${\mathcal{S}}: = \{  X_i \subseteq {\mathbb{R}}^2 \times {\Omega}^s \}$ we define $    \bigcup {\mathcal{S}}:= \bigcup_{X \in {\mathcal{S}}} X        \subseteq {\mathbb{R}}^2 \times \tilde{{\Omega}}^s$.

There are primarily two reasons we have chosen to consider cubical subsets of $ {\mathbb{R}}^2 \times {\Omega}^s$. Firstly, cubes are particularly easy to refine into smaller pieces. This is useful because to begin using a branch and bound method, we need to obtain global bounds on the solution space, and then partition these bounds into smaller pieces. In practice, we reduce the size of a cube by either subdividing it along a lower dimension into two cubes, or replacing the cube by its intersection with the Krawczyk operator: $X \mapsto X \cap K(X,\bar{x})$. In both these cases the 