task. Ideally, one wants some sort of generating function, if it exists, which encodes information about the generators of a basis together with all possible relations among these generators. In addition to allowing us to enumerate the operators in an EFT basis, such a function is appealing since it would encode information about the basis as a whole—it could potentially reveal properties of the EFT that are difficult or impossible to see when working with any fixed subset of operators.

To obtain a function with many of the aforementioned properties, we define a *generalized Hilbert series* as follows. Associate a weight $u_i$ to each field $\ph_i$ in the EFT and a weight $t$ to the (covariant) derivative. The generalized Hilbert series is then defined as $$H(t,\{u_i\}) = \sum_{k=0}^{\infty}\sum_{r_1=0}^{\infty}\dots\sum_{r_N=0}^{\infty} c_{k\, r_1 \dots r_N} t^ku_1^{r_1}\dots u_N^{r_N},
\label{eqn:Hdef}$$ where $c_{k\, r_1 \dots r_N} \in {\mathbb N}$ is the number of independent operators in the basis composed of $k$ derivatives and $r_1,\dots,r_N$ powers of the $\ph_1, \dots, \ph_N$. The weights $\{t,u_i\}$ are complex numbers; formally, in order to have the above series converge, we require $\abs{t},\abs{u_i}< 1$.

The summed Hilbert series succinctly encodes information about the EFT basis. For simplicity, let us assume all 