Partitioning Theorems for Sets of Semi-Pfaffian Sets, with Applications

We generalize the seminal polynomial partitioning theorems of Guth and Katz to a set of semi-Pfaffian sets. Specifically, given a set $Γ\subseteq \mathbb{R}^n$ of $k$-dimensional semi-Pfaffian sets, where each $γ\in Γ$ is defined by a fixed number of Pfaffian functions, and each Pfaffian function is in turn defined with respect to a Pfaffian chain $\vec{q}$ of length $r$, for any $D \ge 1$, we prove the existence of a polynomial $P \in \mathbb{R}[X_1, \ldots, X_n]$ of degree at most $D$ such that each connected component of $\mathbb{R}^n \setminus Z(P)$ intersects at most $\sim \frac{|Γ|}{D^{n - k - r}}$ elements of $Γ$. Also, under some mild conditions on $\vec{q}$, for any $D \ge 1$, we prove the existence of a Pfaffian function $P'$ of degree at most $D$ defined with respect to $\vec{q}$, such that each connected component of $\mathbb{R}^n \setminus Z(P')$ intersects at most $\sim \frac{|Γ|}{D^{n-k}}$ elements of $Γ$. To do so, given a $k$-dimensional semi-Pfaffian set $\mathcal{X} \subseteq \mathbb{R}^n$, and a polynomial $P \in \mathbb{R}[X_1, \ldots, X_n]$ of degree at most $D$, we establish a uniform bound on the number of connected components of $\mathbb{R}^n \setminus Z(P)$ that $\mathcal{X}$ intersects; that is, we prove that the number of connected components of $(\mathbb{R}^n \setminus Z(P)) \cap \mathcal{X}$ is at most $\sim D^{k+r}$. Finally as applications, we derive Pfaffian versions of Szemerédi-Trotter type theorems, and also prove bounds on the number of joints between Pfaffian curves.

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