Khovanskii's Bezout-type Theorem for Pfaffian Functions: A Self-Contained Proof, and Applications

We present a direct and self-contained proof of Khovanskii's Bezout-type bound for the number of nondegenerate solutions of a system of Pfaffian equations. We isolate the ingredients of Khovanskii's original argument and assemble them into a proof that avoids the general theory of integral manifolds developed in his monograph. Our formulation mildly refines the classical statement: rather than depending on the ambient dimension, our bound depends on the maximum number of variables on which any function in the Pfaffian chain depends. As a consequence, we obtain a refined bound on the number of connected components of a Pfaffian set.