Year: 2026
Type: preprint
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 \[ g_σ(z)=1+σ_1 z_1^2+σ_2 z_2^2, \qquad σ_1,σ_2\geq 0. \] 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 \[ 0\leq σ_1,σ_2<1, \qquad \sqrt{1-σ_1^2}+\sqrt{1-σ_2^2}>1. \] 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.
Source: arXiv (Cornell University)
Authors: Danielsen-Jensen, Eden, Grundmeier, Dusty, Helal, Abdullah Al, Kunz, Valentin D., Lebl, Jiri, Xiao, Ming, Zhu, Weixia
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7204914555
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