Parameterwise Sharpness of Khovanskii's Bezout-type Bound for Pfaffian Functions

Khovanskii’s theorem gives a Bezout-type upper bound for the number of isolated real solutions of a system of $n$ Pfaffian equations in $n$ variables in terms of three complexity parameters: the chain-degree $\alpha$, the degrees ${\beta }_i$ of the Pfaffian functions, and the order $s$ of the underlying Pfaffian chain. Despite its fundamental role in Pfaffian geometry and o-minimality, little is known about the sharpness of this bound. We investigate the theorem from a parameter-by-parameter perspective. We show that its dependence on the chain-degree $\alpha$ is asymptotically sharp by constructing, for every $\alpha ,s\in \mathbb{N}$, a Pfaffian function of format $(\alpha ,1,s)$ with at least ${\alpha }^s$ nondegenerate real zeros. We also show that its dependence on the degrees ${\beta }_i$ is asymptotically sharp: for fixed $n$ and $s$, we construct Pfaffian systems having ${\mathrm{\Omega }}_{n,s}({\beta }^{n+s})$ regular common zeros, matching the order of growth predicted by Khovanskii’s theorem as $\beta \to \mathrm{\infty }$.