2024/12/04 by Lotz, Martin, Natarajan, Abhiram, Vorobjov, Nicolai · 2 citations
#Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics #Logic (math.LO)
paper · doi:10.48550/arxiv.2412.02961
We generalize the seminal polynomial partitioning theorems of Guth and Katz to a set of semi-Pfaffian sets. Specifically, given a set Γ⊆ ℝn of k-dimensional semi-Pfaffian sets, where each γ∈ Γ is defined by a fixed number of Pfaffian functions, and each Pfaffian function is in turn defined with respect to a Pfaffian chain q of length r, for any D ≥ 1, we prove the existence of a polynomial P ∈ ℝ[X1, …, Xn] of degree at most D such that each connected component of ℝn ∖ Z(P) intersects at most ∼ \frac|Γ|Dn - k - r elements of Γ. Also, under some mild conditions on q, for any D ≥ 1, we prove the existence of a Pfaffian function P' of degree at most D defined with respect to q, such that each connected component of ℝn ∖ Z(P') intersects at most ∼ \frac|Γ|Dn-k elements of Γ. To do so, given a k-dimensional semi-Pfaffian set X ⊆ ℝn, and a polynomial P ∈ ℝ[X1, …, Xn] of degree at most D, we establish a uniform bound on the number of connected components of ℝn ∖ Z(P) that X intersects; that is, we prove that the number of connected components of (ℝn ∖ Z(P)) ∩ X is at most ∼ Dk+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.