Sharp Region Classifies Degree-Three Rational Sphere Map Denominators

Rational sphere maps are constrained by a boundary norm identity, but even after normalizing the map, it is not apparent which quadratic denominators can occur. The paper resolves that question for degree-three maps from the two-dimensional complex sphere. It studies denominators of the form $g_\sigma(z)=1+\sigma_1z_1^2+\sigma_2z_2^2$ with nonnegative parameters and replaces a nonlinear realization problem with positivity and rank conditions on a finite Gram matrix.

Core Contribution

The central theorem gives a sharp characterization: $g_\sigma$ is the denominator of a normalized degree-three rational sphere map exactly when

This is stronger than constructing examples in a subregion. The strict curved inequality is both necessary and sufficient, so it identifies the boundary beyond which the denominator cannot be realized. The authors also classify the target dimensions available for a fixed admissible denominator and provide Gram normal forms for the corresponding numerators.

The result sits within the classification of proper holomorphic maps between balls. Earlier normal-form results constrain the possible maps, but the denominator parameters still encode a concrete existence problem. Here, the denominator is no longer treated as a by-product of a map construction: it becomes the object classified directly.

Technical Approach

For a rational map $P/g_\sigma$, the sphere-mapping condition is expressed as an identity between Hermitian forms. The authors organize the homogeneous pieces of the numerator in fixed monomial-vector spaces and encode the remaining freedom in a Gram matrix. Positivity of that matrix is the condition that permits a factorization into actual numerator components.

A realization lemma reduces target dimension to a rank calculation of the form

where $M_X$ is the residual cubic Gram matrix, $K$ records the quadratic contribution, and $k$ counts added independent quadratic components. This separation is useful: changing $K$ changes the residual rank, while choosing a nonzero quadratic vector can increase the target dimension without altering that rank.

The proof of admissibility analyzes $M_X$ for a positive diagonal matrix $X$. Its structure splits into two $2\times2$ blocks, allowing the authors to turn positivity into explicit conditions on the two denominator parameters. A rank-one perturbation lemma then identifies when a block remains positive definite and when it loses exactly one rank. That is the mechanism behind the dimension classification, rather than an ad hoc list of numerator formulas.

The paper also gives a scalar full-rank normal form. For every admissible denominator, Corollary 4.2 constructs a reduced normalized map with nine scalar components, showing that the ambient nine-dimensional cubic space is attainable. At $(\sigma_1,\sigma_2)=(0,0)$, the construction specializes to a polynomial sphere map built from weighted linear, quadratic, and cubic monomials.

Results and Analysis

For every admissible nonzero parameter pair, the residual Gram matrix can be made to have ranks 2, 3, or 4. With no added quadratic components, these yield target dimensions 4, 5, and 6. Starting from a strictly positive residual matrix, one or two independent quadratic components produce dimensions 7 and 8; the full-rank construction gives dimension 9. The paper rules out the remaining possibilities using the nine-dimensional cubic numerator space, rigidity results excluding targets 1 and 2, and Faran's classification for the exceptional target-3 case.

The resulting classification is sharp: nonconstant admissible denominators occur in target dimensions 4 through 9, while the polynomial denominator $g\equiv1$ also permits target dimension 3. The rank argument is especially persuasive because it explains why the set is discrete and why dimension 9 is the ceiling, instead of merely exhibiting maps at selected dimensions.

The final extension moves beyond two source variables. The same Gram-matrix framework supplies a general sufficient condition for degree-three rational sphere maps in arbitrary source dimension. That extension broadens the method, but it is not presented as an exact denominator-region classification comparable to the two-variable theorem. The paper's strongest contribution remains the complete low-dimensional result: an explicit semialgebraic region, constructive normal forms, and an exact target-dimension account derived from the same positivity calculation.