Pfaffian Intersections and Multiplicity Cycles

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.