Proper holomorphic maps between balls of different dimensions (Q1106368)

The proper holomorphic mappings from the unit disc to itself are exactly the finite Blaschke products. On the unit ball \(B_ n\) in \({\mathbb{C}}^ n \)with \(n\geq 2\), any proper holomorphic self map is actually biholomorphic, i.e. an automorphism. It remains to classify the proper holomorphic maps from \(B_ n\) to \(B_ m\) for \(m>n\) (there aren't any for \(m<n).\) The present paper contains several results about polynomial proper maps. A one-parameter family of inequivalent proper maps from the n-ball to the 2n-ball is also given. The author then formulates a conjecture concerning ``factorization'' of proper maps from \(B_ n\) to \(B_ m\) into simple ones; this conjecture casts many known results on proper maps between balls into one framework. Finally, a list is given of all the monomial maps from the 2-ball to the 4-ball.