There are references on the exponential worst-case of the intersection operator for regular expressions (see [1]). However, I was wondering if there are similar results for the construction process itself?
Concretely, given two regular expressions $E_1$ and $E_2$ whose intersection can be represented by a regular expression of polynomial size, is the construction process itself bounded by exponential time or memory in the worst-case?
[1] Succinctness of the Complement and Intersection of Regular Expressions