Decidability of Strict Inequality or Equality for \u03b7_i(A) and Attainment of Extrema

Determine whether it is decidable, for a given real-algebraic matrix A and real threshold c, that \u03b7_i(A) > c or \u03b7_i(A) = c; further, determine whether the sequences \mu_n^{(i)}(A) attain their infimum or supremum at some finite index n.

Background

The quantities \u03b7_i(A) and \gamma_i(A) are defined from peripheral spectra of A and track extremal behavior of normalized traces along powers. The authors give decision procedures for non-strict inequalities but highlight a gap for strict inequalities and equality cases.

They trace the difficulty to uncertainty about whether the relevant sequences attain their extremal values at finite indices, which would enable reduction to finite verification.