Positivity and Asymptotics for Chenevier's Orthogonal Polynomials

Abstract

We prove the strict positivity conjectured by Chenevier for the critical vectors in the unconditional part of his automorphic Hermite--Minkowski theorem. The proof establishes strict negativity of all Verblunsky coefficients of a circle measure associated with the weight $(\arcsin x)/x$ on $(-1,1)$. A positive-kernel formula and the classical Schur algorithm give these signs, and a para-orthogonal transformation yields positivity in every degree. We also compute the positive density representing the negative of the Schur function as a Hausdorff moment generating function. After rescaling their indices to $[0,1]$, the normalized critical vectors converge weakly to the arcsine law, while Chenevier's critical scale is asymptotic to $8\pi/n$. At the critical boundary, a single nonzero effective integral vector is negative for every admissible test function exactly in odd degree and in degree zero. Finally, we prove an exact first-variation formula for exponential perturbations of the Legendre measure and derive its asymptotics for endpoint cusps. For the perturbation leading to Chenevier's weight, the derivative at the Legendre measure has a $(\log n)/(\pi^2n^2)$ term and an explicit constant at order $n^{-2}$. The corresponding nonlinear asymptotic remains conjectural.