Upper bounds for degrees of proper holomorphic sphere maps

Determine, for fixed source dimension N and target dimension n+1, an upper bound on the degree of proper holomorphic maps F from the unit ball in C^N to the unit ball in C^{n+1} that map the unit sphere to the unit sphere.

Background

The paper begins by situating one-dimensional discrete models within the study of proper holomorphic maps between complex unit balls. For fixed source and target dimensions, bounding the degree of such maps is identified as a major open problem in Cauchy–Riemann geometry. The paper addresses a special monomial and one-dimensional statistical setting, but does not resolve the general problem for arbitrary proper holomorphic maps.

References

A major open problem is to determine, for fixed $N$ and $n$, an upper bound on the degree of $F$.

— One-dimensional Discrete Models of Maximum Likelihood Degree One  (2507.18686 - Améndola et al., 24 Jul 2025) in Section 1, “From CR Geometry to Algebraic Statistics”