amongst the generators, and therefore the Hilbert series is $$H_{N,\text{EOM}}(t,\ui) = \frac{1}{\prod_{i=1}^N(1-u_i)(1-tu_i)}.
\label{eqn:H_EOM}$$

In the language of section \[sec:framework\], at fixed $r_1,\dots,r_N$ we study the module $$R_{\br}^{G_{\br}}/I_{\br} = \bigslant{\mathbb{R}[P_1^{(1)},\dots,P_1^{(N)}]}{\big\langle \big(P_1^{(1)}\big)^{r_1+1},\dots,\big(P_1^{(N)}\big)^{r_N+1}}\big\rangle,$$ whose Hilbert series is given by $$H_{\br,\text{EOM}}(R_{\br}^{G_{\br}}/I_{\br},t) = \sum_{k=0}^{\infty}t^k \sum_{k_1+\dots+k_N = k} I_{\{k_i \le r_i\}}.
\label{eqn:H_r_EOM}$$ Here we have adopted a notation $I_A$, whose value is $1$ if the condition $A$ is satisfied, and $0$ otherwise. It is also understood that $i$ runs through $1,\cdots,N$ in the set of conditions $\left\{k_i \le r_i\right\}$. The EOM imply the constraints $k_i\le r_i$ and are directly seen from the $\big(P_1^{(i)}\big)^{r_i+1}$ terms in the ideal. We can geometrically sum the above to obtain $$H_{\br,\text{EOM}}(R_{\br}^{G_{\br}}/I_{\br},t) = \sum_{k_1=0}^{r_1}\dots \sum_{k_N=0}^{r_N} t^{k_1+\dots+k_N}= \frac{\prod_{i=1}^{N}(1-t^{r_i+1})}{(1-t)^N}.
\label{eqn:H_r_EOM_sum}$$ We note that the reason why this sum was simple is algebraically rooted in the fact that each term in the ideal depends only on a single indeterminate $P_1^{(i)}$. Upon summing, we reproduce eq. : $$\begin{aligned}
H_{N,\text{EOM}}(t,\ui) &= \sum_{r_i\ge0}\sum_{k\ge0}\sum_{k_1+ \dots + k_N = k} u_1^{r_1}\dots u_N^{r_N} t^k I_{\{k_i\le r_i\}} \nonumber \\
&= \sum_{r'_i\ge 0}u_1^{r'_1}\dots u_N^{r'_N}\sum_{k_i\ge 0} (tu_1^{})^{k_1}\dots (tu_N^{})^{k_N} \nonumber \\
&= \frac{1}{\prod_{i=1}^N(1-u_i)(1-tu_i)},\end{aligned}$$ where in the first to second line we defined $r_i' = r_i-k_i$. We have explicitly written out this step to highlight a 