Finiteness of spectral networks and bounds on noncritical tripods for high-degree polynomial cubic differentials

Determine whether the spectral networks of polynomial cubic differentials of the form \(\varphi=P(x)\,dx^{\otimes 3}\) on \(\mathbf{P}^1\) with large degree \(d\) are finite (i.e., consist of finitely many trajectories), and, contingent on finiteness, ascertain upper bounds on the number of noncritical tripods formed by trajectories starting at intersection points that are not zeros of \(\varphi\).

Background

In the paper the authors study spectral networks associated to cubic differentials on $\mathbf{P}^1$, and obtain complete descriptions for degrees $d\leq 3$. They prove finiteness results and explicit upper bounds for critical tripods (tripods formed by three critical trajectories emanating from zeros).

However, beyond the critical case, the existence and count of tripods formed by trajectories starting at intersection points not located at zeros (noncritical tripods) depend on the global finiteness of the spectral network. The authors note that while noncritical tripods do not appear for $d\leq 3$, for larger degrees it is uncertain whether spectral networks remain finite, and consequently whether one can derive bounds on the number of such tripods.