The degree and generic simple-root conjecture for the Heun spectral polynomial

Let $Q(E)$ be the polynomial whose zeros characterize those values of $E$ for which the equation in the preceding theorem has a solution in the space ${\mathcal F}$, and let $\tilde{V}$ be the maximum finite-dimensional invariant subspace of ${\mathcal F}$ described by the decomposition into spaces $\tilde{V}_{\epsilon_0,\epsilon_1,\epsilon_2,\epsilon_3}$. Let $\omega_1$ and $\omega_3$ be the parameters of the underlying elliptic system. Degree and generic simple-root conjecture. (i) The degree of the polynomial $Q(E)$ in $E$ is equal to the dimension of the space $\tilde{V}$. (ii) For generic $\omega_1$ and $\omega_3$, the roots of the equation $Q(E)=0$ are distinct. These assertions relate the finite-dimensional invariant-subspace structure to the Heun spectral polynomial and predict a simple spectrum for generic elliptic parameters.