In this note we compute the Bergman kernel of the unit ball with respect to the smallest norm in ${ℂ}^n$ that extends the euclidean norm in ${ℝ}^n$ and give some applications.

We describe the set of points over which a dominant polynomial map $f=(f_1,...,f_n):{ℂ}^n\to {ℂ}^n$ is not a local analytic covering. We show that this set is either empty or it is a uniruled hypersurface of degree bounded by $({\prod }_{i=1}^{n}deg{f}_i-\mu \left(f\right))/\left(mi{n}_{i=1,...,n}deg{f}_i\right)$.

We prove that the symmetrized polydisc cannot be exhausted by domains biholomorphic to convex domains.