Amplitude maximization in stable systems, Schur positivity, and some conjectures on polynomial interpolation

For $r>0$ and integers $t\ge n>0$, we consider the following problem: maximize the amplitude $\mathrm{\mid }{x}_t\mathrm{\mid }$ at time $t$, over all complex solutions $x=(x_0,x_1,\dots )$ of arbitrary homogeneous linear difference equations of order $n$ with the characteristic roots in the disc $\{z\in \mathbb{C}:\mathrm{\mid }z\mathrm{\mid }\le r\}$, and with initial values $x_0,\dots ,x_{n-1}$ in the unit disc. We find that for any triple $t,n,r$, the maximum is attained with coinciding roots on the boundary circle; in particular, this implies that the peak amplitude ${\sup }_{t\ge n}\mathrm{\mid }{x}_t\mathrm{\mid }$ can be maximized explicitly, by studying a unique equation with the characteristic polynomial $(z-r{)}^n$. Moreover, the optimality of the cophase root configuration holds for origin-centered polydiscs. To prove this result, we first reduce the problem to a certain interpolation problem over monomials, then solve the latter by leveraging the theory of symmetric functions and identifying the associated Schur positivity structure. We also discuss the implications for more general Reinhardt domains. Finally, we study the problem of estimating the derivatives of a real entire function from its values at $n\mathrm{/}2$ pairs of complex conjugate points in the unit disc. We propose conjectures on the extremality of the monomial $z^n$, and restate them in terms of Schur polynomials.