a given module (even though this might not make sense if the base field $k$ is infinite, since a module may have infinitely many submodules). This theory originates from [@CC06], although $F$-polynomials of modules appeared later in [@DWZ09]. The general results in this section can be found in [@DG12 Section 2].

In the rest of this section, the base field $k$ is the field ${\mathbb{C}}$ of complex numbers.

\[defi::F-polynomial\] Let $V$ be an $A$-module. Its *$F$-polynomial* is $$F_V({\mathbf{y}}) := \sum_{{\mathbf{e}}\in{\mathbb{N}}^{Q_0}} \chi\big( {\mbox{{\rm Gr}}_{{\mathbf{e}}}}(V) \big) {\mathbf{y}}^{\mathbf{e}},$$ where

-   ${\mathbf{y}}$ is the tuple of variables $(y_i \ | \ i\in Q_0)$;

-   ${\mathbf{y}}^{\mathbf{e}}= \prod_{i\in Q_0} y_i^{e_i}$;

-   ${\mbox{{\rm Gr}}_{{\mathbf{e}}}}(V)$ is the submodule Grassmannian (see Definition \[defi::grassmannian\]); and

-   $\chi$ is the Euler-Poincaré characteristic.

We give examples of $F$-polynomials at the end of the section. It is easy to see that the $F$-polynomial of a module $V$ only depends on the isomorphism class of $V$.

\[rema::Euler\] The most difficult part in the computation of an $F$-polynomial is determining the submodule Grassmannians ${\mbox{{\rm Gr}}_{{\mathbf{e}}}}(V)$. To compute their Euler-Poincaré characteristic, the following facts (true since we work over ${\mathbb{C}}$!) are often sufficient (and indeed, suffice to prove all the formulas in these notes):

1.  $\chi({\rm 