Vertical decomposition of shallow levels in 3-dimensional arrangements and its applications

Let ${\cal F}$ be a collection of $n$ bivariate algebraic functions of constant maximum degree. We show that the combinatorial complexity of the vertical decomposition of the ${\le}k$-level of the arrangement ${\cal A}({\cal F})$ is $O(k^{3+\varepsilon}\psi({n/k}))$, for any $\varepsilon