Degree of a holomorphic map between unit balls from $\mathbb C^2$ to $\mathbb C^n$

Abstract:

Let $f$ be a rational proper holomorphic map between the unit ball in $\mathbb C^2$ and the unit ball in $\mathbb C^n.$ Write \[ f=\dfrac {(p_1, \dots , p_n)}{q},\] where $p_j, \ j=1, \dots ,n,$ and $q$ are holomorphic polynomials, with $(p_1,\dots ,p_{n},q)$ $=1.$ Recall that the degree of $f$ is defined by \[ \text {deg} f =\text {max}\{\text {deg} (p_j)_{j=1,\dots ,n}, \text {deg} q\}.\] In this paper, we give a bound estimate for the degree of $f,$ improving the bound given by Forstnerič (1989).