CHEBYSHEV–LUCAS CRITICAL POLYNOMIALS: LOCAL BLOCKS, IRREDUCIBILITY AND GALOIS GROUPS

Abstract

e introduce and study a family of critical polynomials arising naturally from the maximization problem for the sharp characteristic-function dependence constant \(C_d\). The critical equation for the optimizer admits equivalent Chebyshev and Lucas formulations and leads to a distinguished polynomial family whose roots encode the algebraic structure of the extremal point. We develop the arithmetic theory of these polynomials, including parity-dependent reductions, irreducibility criteria, \(p\)-adic Newton polygons, local factorization and ramification, primitive-divisor obstructions to low-degree factors, and Galois-group results. Irreducibility is established for broad infinite arithmetic families and is computationally certified through \(3\le d\le400\). A principal consequence is that, whenever the critical polynomial is irreducible, \[ [\mathbb Q(C_d):\mathbb Q] = \begin{cases} d-1,& d\ \text{odd},\\ d/2-1,& d\ \text{even}, \end{cases} \] with unconditional instances supplied by the proved irreducibility classes. Thus an extremal problem originating in probability and characteristic functions leads naturally to a local-global arithmetic problem involving Lucas sequences, Newton polygons, and large Galois groups.