Describe the boundary of the minimal T_{≥n}-invariant set for non-degenerate and certain degenerate operators

Determine the boundary of the minimal closed convex T_{≥n}-invariant set M_{≥n} for linear differential operators T with polynomial coefficients, in the cases where T is non-degenerate or T is degenerate with a non-defining Newton polygon and with nonconstant leading coefficient Q_k(x).

Background

For a differential operator T = \sum_{j=0}^k Q_j(x) d^j/dx^j with nonconstant leading coefficient Q_k, the paper defines, for each n ≥ 0, the class of closed sets invariant under T on polynomials of degree at least n, and shows there is a unique minimal element M_{≥n}. For non-degenerate T, large n yield compact minimal sets and M_∞ equals the fundamental polygon Conv(Q_k).

Despite these structural results and asymptotic descriptions (e.g., existence of invariant disks for large n and the limit M_∞ = Conv(Q_k)), the precise boundary of M_{≥n} is not known in general, even for order-1 non-degenerate operators.