  We consider the problem of estimating the intersection multiplicity between
an algebraic variety and a Pfaffian foliation, at every point of the variety.
We show that this multiplicity can be majorized at every point $p$ by the local
algebraic multiplicity at $p$ of a suitably constructed algebraic cycle. The
construction is based on Gabrielov's complex analog of the Rolle-Khovanskii
lemma.
  We illustrate the main result by deriving similar uniform estimates for the
complexity of the Milnor fiber of a deformation (under a smoothness assumption)
and for the order of contact between an algebraic hypersurface and an arbitrary
non-singular one-dimensional foliation. We also use the main result to give an
alternative geometric proof for a classical multiplicity estimate in the
context of commutative group varieties.
