Mundici considered the question of whether the interpolant of two
propositional formulas of the form $F\rightarrow G$ can always have
a short circuit description, and showed that if this is the case then
every problem in NP $\cap$ co-NP would have polynomial size circuits.
In this note we observe further consequences of the interpolant having
short circuit descriptions, namely that
UP $\subseteq$ P$/$poly, and that every single valued NP function has a
total extension in FP$/$poly. We also relate
this question with other
Complexity Theory assumptions.