of its generalizations to EFTs in higher space-time dimensions.

We include a basic introduction to commutative algebra in appendix \[app:a\] and a demonstration implementing our framework in the computer package [Macaulay2]{} in appendix \[app:b\].

Framework {#sec:framework}
=========

In this section we provide a systematic framework for computing the Hilbert series, *i.e.* counting operators modulo integration by parts and use of the equations of motion. Throughout this section, we restrict the discussion to our one-dimensional QFT with $N$ flavors of real scalar fields $\phi_i$, $i=1,\dots,N$. We take the spacetime manifold to be the circle $S^1$ with coordinate $\th$; the counting is the same here as for the real line.

We take the equations of motion from $\int_{S^1}d\th \sum_i (\pd\ph_i)^2/2$, which gives $\pd^2\ph_i=0$. In one-dimension, terms of the form $\ph_j\pd \ph_i$, $i\ne j$, are present in the action and could be included in the EOM. However, they do not change the counting because $\pd^2\ph_i$ is always in the EOM as long as $\phi_i$ has a non-zero kinetic term.

Introducing a shorthand notation $\bu^{\br}$ for $u_1^{r_1}\dots u_N^{r_N}$ and $c_{k\, \br} = c_{k\, r_1\dots r_N}$, the Hilbert series is written as $$H_N(t,\ui) = \sum_{k,\mathbf{r}=0}^{\infty}c_{k\, \mathbf{r}}\mathbf{u}^{\mathbf{r}} t^k.
\label{eqn:hilbshort}$$ We recall that $c_{k\, \br}$ is the 