Partitioning Theorems for Sets of Semi-Pfaffian Sets, with Applications
Martin Lotz, Abhiram Natarajan, Nicolai Vorobjov
TL;DR
The paper develops polynomial and Pfaffian partitioning tools for finite families of semi-Pfaffian sets in Euclidean space, extending Guth–Katz partitioning to the o-minimal Pfaffian setting. It proves a polynomial partitioning theorem with a degree-$D$ polynomial P that evenly distributes intersections across components, and a parallel Pfaffian partitioning theorem using a Pfaffian function P' of degree at most $D$ under an algebraic-independence condition. A key technical contribution is a semi-Pfaffian Barone–Basu–type bound and a smooth decomposition technique that control the complexity and topology of intersections, enabling Szemerédi–Trotter type incidence theorems and joints bounds for Pfaffian curves. The results provide Pfaffian analogues of classical incidence bounds, with applications to point–Pfaffian-curve incidences and joints in $ ext{R}^n$, and highlight both the potential and challenges of extending algebraic partitioning techniques to broader o-minimal definable families.
Abstract
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.