Characterization of algebraic curves that satisfy the Phragmen-Lindeloef principle for global evolution

For a complex polynomial P in two distinguished variables it is characterized when its zero variety V(P) satisfies the Phragmen-Lindeloef condition that is equivalent to the differential operator P(D) being of evolution on spaces of ultradifferentiable functions. The methods that are used also apply to certain systems of such operators as some examples show