Degree-Three Rational Sphere Maps: Sharp Denominator Region and Gram Normal Forms
We study degree-three rational sphere maps in two complex variables. After a standard normalization, the denominator of such a map takes the form \[ g_\sigma(z)=1+\sigma_1 z_1^2+\sigma_2 z_2^2, \qquad \sigma_1,\sigma_2\geq 0. \] A basic question is: which pairs $(\sigma_1,\sigma_2)$ 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 \[ 0\leq \sigma_1,\sigma_2<1, \qquad \sqrt{1-\sigma_1^2}+\sqrt{1-\sigma_2^2}>1. \] Our approach converts the sphere-mapping condition into a finite-dimensional Gram-matrix positivity problem. Furthermore, for each admissible parameter $ \sigma=(\sigma_1,\sigma_2), $ we determine all possible minimal target dimensions in which the corresponding denominator $g_\sigma$ can be realized. We also give a Gram-matrix normal form for maps with a fixed denominator and compute, for each admissible $\sigma$, the dimension of the moduli space of equivalence classes of rational sphere maps realizing $g_\sigma$. 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.