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 8π/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)/(π2n2) term and an explicit constant at order n−2. The corresponding nonlinear asymptotic remains conjectural.
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BibTeX
@misc{xu2026positivity,
title = {{Positivity and Asymptotics for Chenevier's Orthogonal Polynomials}},
author = {Shisong Xu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10328},
eprint = {2609.10328},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10328},
}
RIS
TY - UNPB
TI - Positivity and Asymptotics for Chenevier's Orthogonal Polynomials
AU - Shisong Xu
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.10328
UR - https://arxiv.org/abs/2609.10328
AB - 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 8π/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)/(π^2n^2) term and an explicit constant at order n^−2. The corresponding nonlinear asymptotic remains conjectural.
ER -