the action as, $$\begin{aligned}
\int_{S_1} d\th \, \ph_1^{r_1}\dots \ph_{N}^{r_N} \pd^k = \sum_{\{q^{(1)}\}\dots \{q^{(N)}\} = -\infty}^{\infty} & w\big(\{q^{(1)}\}\dots \{q^{(N)}\}\big) \times \big(a_1^{(1)}\dots a_{r_1}^{(1)}\big) \dots \big(a_1^{(N)}\dots a_{r_N}^{(N)}\big) \nonumber \\
&\times \int_{S_1} d\th \exp\left( i \th \sum_{i=1}^N\sum_{j=1}^{r_i}q_j^{(i)}\right),\end{aligned}$$ where $w$ is a degree $k$ polynomial of the momenta, invariant under the symmetric group $S_{r_i}$ for each set of momenta $\{q^{(i)}\}$. We denote this ring by $R_{\mathbf{r}}^{G_{\br}} = \mathbb{R}[\{q^{(1)}\},\dots,\{q^{(N)}\}]^{G_{\br}}$ where $G_{\br} = S_{r_1} \times \dots \times S_{r_N}$. $R_{\br}^{G_{\br}}$ is freely generated by the power sum symmetric polynomials $P_1^{(i)},\dots,P_{r_i}^{(i)}$: $R_{\br}^{G_{\br}} = \mathbb{R}[\{P^{(1)}\},\dots,\{P^{(N)}\}]$.

As in eq. , IBP is handled by momentum conservation, $P_1^{(1)} + \dots + P_1^{(N)} = 0$, while the EOM imply $P_n^{(i)} = 0$ for $n \ge 2$ as well as $\big(P_1^{(i)}\big)^{r_i+1} = 0$. Together, these equations form an ideal of $R_{\br}^{G_{\br}}$. The quotient ring containing the equivalence classes of operators is $$\bigslant{R_{\br}^{G_{\br}}}{I_{\br}} = \bigslant{\mathbb{R}[P_1^{(1)},\dots,P_1^{(N)}]}{\big\langle P_1^{(1)}+\dots + P_1^{(N)},\big(P_1^{(1)}\big)^{r_1+1},\dots,\big(P_1^{(N)}\big)^{r_N+1}}\big\rangle.
\label{eqn:module}$$

Questions about the EFT basis can now be studied by examining the structure of the modules in eq. . One of our main interests is determining the number of operators in the EFT basis, and this information is encoded in the Hilbert series $H_{\br}(R_{\br}^{G_{\br}}/I_{\br},t)$. Computing $H_{\br}(R_{\br}^{G_{\br}}/I_{\br},t)$ for arbitrary $\br$ and $N$ is fairly involved. However, $H_{\br}(R_{\br}^{G_{\br}}/I_{\br},t)$ 