The characteristic frequencies of a linear, shift-invariant multidimensional behavior correspond to its nonzero exponential trajectories. The set of polynomial-exponential trajectories belonging to a fixed characteristic frequency of a behavior is investigated: A test is derived for determining whether this space is finite-dimensional, and if so, a basis is constructed. If it is infinite-dimensional, one considers only the polynomial-exponential trajectories up to a certain degree of the polynomial part, and a characterization is given of the asymptotic growth of the dimensions of these spaces as the degree bound tends to infinity. A dual problem is concerned with linear exact modeling, that is, the construction of the so-called most powerful unfalsified model (MPUM): Given a finite set of polynomial-exponential trajectories, the goal is to construct a behavior that contains the data and as little else as possible.