can always get rid of all derivatives acting on a certain field $\ph_i$, reallocating all of the $k$ derivatives onto the other fields. From the ideal, this is seen by using the momentum conservation equation $\sum_{i=1}^NP_1^{(i)} = 0$ to eliminate a chosen $P_1^{(i)}$.[^8] It is then clear that if $k$ is greater than $r-r_i$ for any $i \in \{1,\cdots,N\}$, the operator would be zero. Therefore, we also have $$c_{k\,\textbf{r}}^{} \propto I_{\left\{ {k \le r - {r_i}} \right\}} .$$

Combining these constraints, the Hilbert series is given by $$H = \sum\limits_{\br=0}^\infty  \bu^\br \sum\limits_{k = 0}^\infty  t^k I_{\left\{ {k \le r - {r_i}} \right\}}I_{2k \le r}\left[ \sum\limits_{{k_1} +  \cdots  + {k_N} = k} I_{\left\{ {{k_i} \le {r_i}} \right\}} - \sum\limits_{{k_1} +  \cdots  + {k_N} = k - 1} I_{\left\{ {{k_i} \le {r_i}} \right\}} \right] . \label{eqn:HFinal_Sum}$$ Note that the conditions $I_{\{k \le r - r_i\}}$ and $I_{2k \le r}$ have overlap—which one dominates depends on $r_1,\dots,r_N$. Let $r_j$ be the maximum number in a given $r_1,\dots,r_N$, $r_j \ge r_i$ for all $i$. The set of conditions $\{k \le r -r_i\}$ is then equivalent to the single condition $k \le r - r_j$. If $r_j \ge \sum_{i\ne j}r_i$, then the condition $2k \le 