degree $j$, $I = \avg{f}$ with $\text{deg}(f) = j$, an elementary calculation tells us that the number of independent degree $k$ polynomials in $R/I$ is equal to the number at degree $k$ in $R$ minus the number at degree $k-j$ in $R$ (see appendix \[app:a\]). Therefore, since $P_{1}^{(1)} + \dots + P_{1}^{(N)}$ is homogeneous and of degree one, the Hilbert series of the quotient ring is given by $$\begin{aligned}
H_{\br,\text{IBP}}(R_{\br}^{S_{\br}}/I_{\br};t) &= \sum_{k=0}^{\infty}t^k \left[\sum_{k_1+\dots+k_N = k} - \sum_{k_1+\dots+k_N = k-1}\right] \\
&= \frac{1-t}{\prod_{i=1}^N\prod_{n_i=1}^{r_i}(1-t^{n_i})},\end{aligned}$$ where the second equality holds for $\br \ne 0$ and is unity for $\br = 0$. Each set $\{P^{(i)}\}$ contributes to the above Hilbert series analogous to the single set in eq. ; the $1-t$ in the numerator reflects the fact that the ideal essentially removes one of the $P_1^{(i)}$. Summing $\sum_{\br} \mathbf{u}^{\br}H_{\br}(t)$ we obtain the full Hilbert series as in eq. .

Only relations from equations of motion {#only-relations-from-equations-of-motion .unnumbered}
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The equations of motion are $\pd^2\ph_i = 0$; consequently $\pd^k\ph_i = 0$ for $k\ge 2$. Thus, when including only the relations from equations of motion, all operators are generated by the set $\{\ph_i,\pd\ph_i\}$. Note that the EFT basis is finitely generated. There are no non-trivial relations 