The degree and ramification conjecture for Picard–Fuchs operators of maximally mutable Laurent polynomials

Let $P$ be a Fano polygon with singularity content $(k,\{ n\times\frac13(1,1)\})$. Let $f$ be a maximally mutable Laurent polynomial with $\operatorname{Newt}(f)=P$, and let $L_f$ be its associated Picard–Fuchs operator. Write $g=g_{mut}$ for the mutation genus, and let $rf(L_f)$ denote the ramification index of the local system associated to $L_f$. Let $k_{eff}$ be the number of multiple points on the curve $f=0$.

Degree and ramification conjecture. The following hold:

1. The degree of $L_f$ is

$$g^2+3g-1+2g\cdot rf(L_f).$$

2. The ramification index is

$$rf(L_f)=n+k_{eff}-3.$$

These formulas propose a relationship between the Picard–Fuchs operator, the mutation genus, the ramification defect, and the singularity content of the Fano polygon. The surrounding discussion presents them as an empirical pattern observed for explicitly computed examples, including smooth Fano polygons and several polygons with singularity content $(k,\{n\times\frac13(1,1)\})$; their general validity is not established in the supplied text.