of $r$ powers of $\ph$ fields and $k$ derivatives decomposes as $$\ph^r\pd^k = \sum_{q_1,\dots,q_r=-\infty}^{\infty} w(q_1,\dots,q_r)a_{q_1}\dots a_{q_r} e^{i(q_1+\dots+q_r)\th},
\label{eqn:op_fourier}$$ where $w$ is a degree $k$ polynomial in the momenta $q_1,\dots,q_r$. One can always symmetrize $w$ due to the permutation symmetry among the dummy indices $q_i$—this is just the fact that the $\phi$’s are indistinguishable. Because a degree $k$ symmetric polynomial uniquely determines an operator with $k$ derivatives, and vice-versa, we can translate the operator counting to that of counting polynomials. This leads us to consider [*polynomial rings*]{} in the momenta; we shall see that the IBPs and EOM imply polynomial constraints, the consequences of which are embodied in [*ideals*]{} of the rings.[^4]

Let $R_r=\mathbb{R}[q_1,\dots,q_r]$ be the polynomial ring in the $r$ momenta with real coefficients. The symmetric polynomials form a subring $R_r^{S_r}= \mathbb{R}[q_1,\dots,q_r]^{S_r}\subset R_r$. It is a well known result, *e.g.* [@Sturmfels:inv], that the ring $R_r^{S_r}$ is freely generated by the power sum symmetric polynomials $P_1, \dots, P_r$ defined as $$P_n = \sum_{s=1}^{r}q_s^n,
\label{eqn:power_sum}$$ and so we have $R_r^{S_r} = \mathbb{R}[P_1,\dots,P_r]$.

What happens when we consider IBP and EOM? In the above exposition, integration by parts manifests as the statement of momentum conservation. When an operator in the action, $\int_{s^1} d\th 