Slope-determinant method, complex cellular structures and hypersurface coverings of regular rational points

2024/12/06 by Kenneth Chung Tak Chiu, Chiu, Kenneth Chung Tak
Computer Science · Engineering · Mathematics · #Advanced Differential Equations and Dynamical Systems #Advanced Numerical Analysis Techniques #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2412.05205

openalex publication_date 2024/12/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We use the determinant method of Bombieri-Pila and Heath-Brown and its Arakelov reformulation by Chen utilizing Bost's slope method to estimate the number of hypersurfaces required to cover the regular rational points with bounded Arakelov height on a projective variety. Using complex cellular structures introduced by Binyamini-Novikov, we replace the usual subpolynomial factor by a polylogarithmic factor in the estimation.

Related

• Determinant method and the pseudo-effective threshold
• Beyond heights: slopes and distribution of rational points
• On the global determinant method
• Counting rational points on cubic hypersurfaces
• The density of rational points on non-singular hypersurfaces, II