Hsiang cubics as spherical Chebyshev polynomials: equioscillation and idempotent quadrature
Abstract
A Hsiang cubic is a cubic form on ℝn whose zero set is a minimal cone; equivalently, it is the cubic form of a commutative nonassociative metrized algebra satisfying a single identity of degree four. We show that the intrinsic geometry of such an algebra, encoded in its idempotents and their Peirce decompositions, is the geometry of a Chebyshev-type extremal problem on the sphere. A Hsiang cubic equioscillates on Sn−1: its maximum set is the set of normalized idempotents, and on suitable great circles through an idempotent it restricts to the Chebyshev polynomial T3. For the spherical analogue of the problem of least deviation, the minimization of |p|∞2/|p|22 over harmonic cubics, a sharp lower bound based on Kellogg's inequality shows that Cartan's isoparametric cubics are the global minimizers in dimensions $5$, $8$, $14$ and $26$. In general, we compute the ratio of every Hsiang cubic in terms of the Killing form and characterize stationarity by a Kolmogorov criterion: it amounts to a positive measure on the idempotent variety. Such measures are given explicitly in several cases which together cover every Hsiang cubic known to exist; in particular, all of them are stationary. These measures give quadrature identities on the idempotent variety whose moments are invariants of the algebra; they are spherical designs exactly in the Einstein (Killing metrized) case, i.e.\ when the Killing form is proportional to the inner product. Distinguished low-dimensional Hsiang cubics are strict local minimizers, whereas a polar Hsiang cubic is a stationary non-minimizer.
Related
- Gauss--Chebyshev Quadrature, 𝕊1-Designs, and the Planar Minkowski Inverse Problem: A One-Dimensional Model for Design-Induced Convex Geometry
- Chebyshev polynomials on a Jordan arc
- Projective symmetries of Chebyshev polynomials
- Chebyshev's method applied to polynomials with rotational symmetry
- Positive Bidiagonal Factorizations and Mixed-Type Chebyshev Multiple Orthogonal Polynomials
BibTeX
@misc{tkachev2026hsiang,
title = {{Hsiang cubics as spherical Chebyshev polynomials: equioscillation and idempotent quadrature}},
author = {Vladimir G. Tkachev},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.13532},
eprint = {2610.13532},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.13532},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Hsiang cubics as spherical Chebyshev polynomials: equioscillation and idempotent quadrature
AU - Vladimir G. Tkachev
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.13532
UR - https://arxiv.org/abs/2610.13532
AB - A Hsiang cubic is a cubic form on ℝ^n whose zero set is a minimal cone; equivalently, it is the cubic form of a commutative nonassociative metrized algebra satisfying a single identity of degree four. We show that the intrinsic geometry of such an algebra, encoded in its idempotents and their Peirce decompositions, is the geometry of a Chebyshev-type extremal problem on the sphere. A Hsiang cubic equioscillates on S^n−1: its maximum set is the set of normalized idempotents, and on suitable great circles through an idempotent it restricts to the Chebyshev polynomial T_3. For the spherical analogue of the problem of least deviation, the minimization of |p|_∞^2/|p|_2^2 over harmonic cubics, a sharp lower bound based on Kellogg's inequality shows that Cartan's isoparametric cubics are the global minimizers in dimensions $5$, $8$, $14$ and $26$. In general, we compute the ratio of every Hsiang cubic in terms of the Killing form and characterize stationarity by a Kolmogorov criterion: it amounts to a positive measure on the idempotent variety. Such measures are given explicitly in several cases which together cover every Hsiang cubic known to exist; in particular, all of them are stationary. These measures give quadrature identities on the idempotent variety whose moments are invariants of the algebra; they are spherical designs exactly in the Einstein (Killing metrized) case, i.e.\ when the Killing form is proportional to the inner product. Distinguished low-dimensional Hsiang cubics are strict local minimizers, whereas a polar Hsiang cubic is a stationary non-minimizer.
ER -