Maximize amplitude of linear recurrence

Write the general solution of a homogeneous linear difference equation in terms of its characteristic roots.

Explain how solutions behave when characteristic roots have multiplicity, leading to terms of the form t^k * r^t.

Formulate the paper's optimization problem: maximizing the solution amplitude |x_t| given constraints on the characteristic roots (|z| ≤ r) and initial conditions.

Explain the Maximum Modulus Principle for analytic functions on compact domains.

Apply the Maximum Modulus Principle to argue that the optimal characteristic roots for maximizing amplitude must lie on the boundary of the disc, i.e., |z| = r.

Explain how the initial values of a difference equation define a polynomial interpolation problem.

Restate the amplitude maximization problem as a problem of maximizing a linear combination of monomial values, subject to interpolation constraints.

Define elementary symmetric polynomials and relate them to the coefficients of a characteristic polynomial.

Describe Schur polynomials as a basis for the space of symmetric functions.

Explain the concept of Schur positivity and its conceptual role in proving that a certain configuration is optimal.

Define 'cophase roots' and explain why a characteristic polynomial like (z-r)^n represents this configuration.

Connect the Schur positivity argument to the conclusion that the cophase root configuration is optimal for the interpolation problem.

State the paper's main result: the maximum amplitude |x_t| is achieved when all characteristic roots are equal to r.

Interpret the result for the peak amplitude sup_{t≥n} |x_t|.

Describe how the main result extends from a disc to origin-centered polydiscs.