Cohn Trace Polynomials: Root Localization, Strict Log-Concavity, and Rearrangement Inequalities

Abstract

We study the one-variable trace polynomials obtained from words in the rank-two free group by the Cohn normalization. We prove that every nontrivial word has all its zeros in a fixed sector about the negative real axis; after the marking involution, this is equivalently a translated exterior-sector statement for the diagonal trace polynomial. The geometric proof uses the three-three Riley slice and its Farey trace recursion, while a factorization and convolution argument converts the root bound into strict coefficient log-concavity for arbitrary words. For positive words, an invariant coefficient cone gives positive integer coefficients,strict domination of proper consecutive subwords, and extremal rearrangement inequalities. The Christoffel-word cases yield coefficientwise monotonicity and the three associated Markov-number inequalities. Finally, commutator powers show that the translation by three is the least real translation with these root and coefficient properties.