Distinct Distances on Pfaffian Curves

2025/10/05 by Abhiram Natarajan, Natarajan, Abhiram, Adam Sheffer +1 · 1 citation
Computer Science · Mathematics · #Combinatorics (math.CO) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #Logic (math.LO) #Mathematical functions and polynomials #Mathematics and Applications #Metric Geometry (math.MG) #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2510.04337

openalex publication_date 2025/10/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We generalize Pach and de Zeeuw's bound for distinct distances between points on two curves, from algebraic curves to Pfaffian curves. Pfaffian curves include those that can be defined by any combination of elementary functions, including exponential and logarithmic functions, rational and irrational powers, trigonometric functions and their inverses, integration, and more. The bound remains Ω(min\m^3/4n^3/4,m²,n²\), as obtained from the proximity technique of Solymosi and Zahl.

Citations

• Partitioning Theorems for Sets of Semi-Pfaffian Sets, with Applications
• Incidence geometry and polynomial expansion over finite fields
• Distinct Distances in R³ Between Quadratic and Orthogonal Curves
• Model-theoretic Elekes-Szabó for stable and o-minimal hypergraphs
• Cutting lemma and Zarankiewicz's problem in distal structures
• Regularity lemma for distal structures
• A semi-algebraic version of Zarankiewicz's problem
• Distinct Distances: Open Problems and Current Bounds
• Polynomials vanishing on grids: The Elekes-Rónyai problem revisited
• Distinct distances on two lines

Cited by

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

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