Maps between \(B^n\) and \(B^N\) with geometric rank \(k_0\leq n-2\) and minimum \(N\)
Let \(\text{Rat}(B^n, B^N)\) the set of rational proper holomorphic maps of \(B^n\) into \(B^N\) \((2\leq n\leq N)\), \(B^N\) denoting the unit ball in \(\mathbb{C}^N\). Using the notion of geometric rank introduced by X. Huang in 2003, the authors prove: Theorem 1.1. Let \(F\in \text{Rat}(B^n, B^N)\) with geometric rank \(\kappa_0\), \(1\leq\kappa_0\leq n-2\), and with \(N= n+{(2n-\kappa_0- 1)\kappa_0\over 2}\). Then \(\deg(F)\leq \kappa_0+ 2\). Theorem 1.2. Let \(F\in \text{Rat}(B^4, B^9)\) with the geometric rank 2 and with \(\deg(F)= 2\). Then \(F\) is equivalent to Whitney map \(W_{4,2}\) of rank 2.
scientific article; zbMATH DE number 2148820