Radial reduction, spherical splitting and root counts for two-sided quaternionic polynomials
Abstract
We study quaternionic polynomials whose coefficients may occur on both sides of the variable, using their degree in the four real coordinates. Such polynomials can have no roots, or many more isolated roots than their degree. A simple root has an invertible real derivative, and its sign is the sign of that derivative's determinant. For a radial family, the roots over a non-real target reduce to one real equation. We obtain a sharp root bound and follow a change in the signed root count as roots escape to infinity. The main results concern spherical root sets for which the polynomial derivative is invertible in the two normal directions. Under a small polynomial perturbation, a sphere splits into simple roots corresponding to the zeros of a tangent vector field, when the derivative of that field is invertible at every zero. We obtain a sharp bound in terms of the perturbation degree and a signed count of two or of minus two. For even degrees, the construction yields polynomials with a single leading power and lower-degree terms whose simple root counts, divided by the fourth power of the degree, tend to one as the degree grows. An explicit quartic has 132 simple roots, and this count persists under small coefficient perturbations. For gradient perturbations, the roots correspond to equilibrium directions of an energy on the sphere. This correspondence gives energy thresholds for root disappearance and bounds the critical directions of polynomial orientation profiles in diffusion magnetic resonance imaging.
Related
- Dual slice functions and dual quaternionic polynomials
- Lengths and incidences in Poisson hypersphere and spherical splitting tessellations
- Splitting sums of binary polynomials
- A 2X2 Matrix Approach to Hopf Fibrations: Quaternionic Reduction of Octonionic Bi-unitary Matrices and an RLLR/LRRL Bracketing Rule
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BibTeX
@misc{laniewski2026radial,
title = {{Radial reduction, spherical splitting and root counts for two-sided quaternionic polynomials}},
author = {R. Laniewski},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33082},
eprint = {2610.33082},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33082},
}
RIS
TY - UNPB
TI - Radial reduction, spherical splitting and root counts for two-sided quaternionic polynomials
AU - R. Laniewski
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.33082
UR - https://arxiv.org/abs/2610.33082
AB - We study quaternionic polynomials whose coefficients may occur on both sides of the variable, using their degree in the four real coordinates. Such polynomials can have no roots, or many more isolated roots than their degree. A simple root has an invertible real derivative, and its sign is the sign of that derivative's determinant. For a radial family, the roots over a non-real target reduce to one real equation. We obtain a sharp root bound and follow a change in the signed root count as roots escape to infinity. The main results concern spherical root sets for which the polynomial derivative is invertible in the two normal directions. Under a small polynomial perturbation, a sphere splits into simple roots corresponding to the zeros of a tangent vector field, when the derivative of that field is invertible at every zero. We obtain a sharp bound in terms of the perturbation degree and a signed count of two or of minus two. For even degrees, the construction yields polynomials with a single leading power and lower-degree terms whose simple root counts, divided by the fourth power of the degree, tend to one as the degree grows. An explicit quartic has 132 simple roots, and this count persists under small coefficient perturbations. For gradient perturbations, the roots correspond to equilibrium directions of an energy on the sphere. This correspondence gives energy thresholds for root disappearance and bounds the critical directions of polynomial orientation profiles in diffusion magnetic resonance imaging.
ER -