Degree-Three Rational Sphere Maps: Sharp Denominator Region and Gram Normal Forms
Abstract
We study degree-three rational sphere maps in two complex variables. After a standard normalization, the denominator of such a map takes the form gσ(z)=1+σ1 z12+σ2 z22, σ1,σ2≥ 0. A basic question is: which pairs (σ1,σ2) can actually occur as the denominator of a degree-three rational sphere map? The first main result of the paper gives a complete answer: such a denominator occurs if and only if 0≤ σ1,σ2<1, √(1−σ12)+√(1−σ22)>1. Our approach converts the sphere-mapping condition into a finite-dimensional Gram-matrix positivity problem. Furthermore, for each admissible parameter $ σ=(σ_1,σ_2), $ we determine all possible minimal target dimensions in which the corresponding denominator gσ can be realized. We also give a Gram-matrix normal form for maps with a fixed denominator and compute, for each admissible σ, the dimension of the moduli space of equivalence classes of rational sphere maps realizing gσ. Finally, we extend the Gram-matrix method to arbitrary source dimension and obtain a general sufficient condition for the existence of degree-three rational sphere maps.
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BibTeX
@misc{danielsenjensen2026degreethree,
title = {{Degree-Three Rational Sphere Maps: Sharp Denominator Region and Gram Normal Forms}},
author = {Eden Danielsen-Jensen and Dusty Grundmeier and Abdullah Al Helal and Valentin D. Kunz and Jiri Lebl and Ming Xiao and Weixia Zhu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.29451},
eprint = {2608.29451},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29451},
}
RIS
TY - UNPB
TI - Degree-Three Rational Sphere Maps: Sharp Denominator Region and Gram Normal Forms
AU - Eden Danielsen-Jensen
AU - Dusty Grundmeier
AU - Abdullah Al Helal
AU - Valentin D. Kunz
AU - Jiri Lebl
AU - Ming Xiao
AU - Weixia Zhu
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2608.29451
UR - https://arxiv.org/abs/2608.29451
AB - We study degree-three rational sphere maps in two complex variables. After a standard normalization, the denominator of such a map takes the form g_σ(z)=1+σ_1 z_1^2+σ_2 z_2^2, σ_1,σ_2≥ 0. A basic question is: which pairs (σ_1,σ_2) can actually occur as the denominator of a degree-three rational sphere map? The first main result of the paper gives a complete answer: such a denominator occurs if and only if 0≤ σ_1,σ_2<1, √(1−σ_1^2)+√(1−σ_2^2)>1. Our approach converts the sphere-mapping condition into a finite-dimensional Gram-matrix positivity problem. Furthermore, for each admissible parameter $ σ=(σ_1,σ_2), $ we determine all possible minimal target dimensions in which the corresponding denominator g_σ can be realized. We also give a Gram-matrix normal form for maps with a fixed denominator and compute, for each admissible σ, the dimension of the moduli space of equivalence classes of rational sphere maps realizing g_σ. Finally, we extend the Gram-matrix method to arbitrary source dimension and obtain a general sufficient condition for the existence of degree-three rational sphere maps.
ER -