Let $G \subset \mathbb{ C}^2$ be a smoothly bounded pseudoconvex domain and assume that the Bergman kernel of $G$ is algebraic of degree $d$. We show that the boundary $\partial G $ is of finite type and the type $r$ satisfies $r\leq 2d$. The inequality is optimal as equality holds for the egg domains $\{ |z|^2+|w|^{ 2s}<1\},$ $s \in \mathbb{ Z}_+$, by D'Angelo's explicit formula for their Bergman kernels. Our results imply, in particular, that a smoothly bounded pseudoconvex domain $G \subset \mathbb{ C}^2$ cannot have rational Bergman kernel unless it is strongly pseudoconvex and biholomorphic to the unit ball by a rational map. Furthermore, we show that if the Bergman kernel of $G$ is rational of the form $\frac{ p}{ q}$, reduced to lowest degrees, then its rational degree $\max\{ \text{ deg } p, \text{ deg } q \}\geq 6$. Equality is achieved if and only if $G$ is biholomorphic to the unit ball by a complex affine transformation of $\mathbb{ C}^2$.
Categories: math.CV math.DG