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 $$g_{\sigma }(z)=1+{\sigma }_1{z}_1^2+{\sigma }_2{z}_2^2,{\sigma }_1,{\sigma }_2\ge 0.$$ 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 $$0\le {\sigma }_1,{\sigma }_2<1,\sqrt{1-{\sigma }_1^2}+\sqrt{1-{\sigma }_2^2}>1.$$ 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.