Dimension growth for affine varieties

2023/11/09 by Floris Vermeulen, Vermeulen, Floris · 3 citations
Computer Science · Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Meromorphic and Entire Functions #Number Theory (math.NT) #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2311.05433

openalex publication_date 2023/11/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove uniform upper bounds on the number of integral points of bounded height on affine varieties. If X is an irreducible affine variety of degree d≥ 4 in \mathbbAⁿ which is not the preimage of a curve under a linear map \mathbbAⁿ→ \mathbbA^{n-dim X+1}, then we prove that X has at most O_d,n,ε(B^{dim X - 1 + ε}) integral points up to height B. This is a strong analogue of dimension growth for projective varieties, and improves upon a theorem due to Pila, and a theorem due to Browning-Heath-Brown-Salberger. Our techniques follow the p-adic determinant method, in the spirit of Heath-Brown, but with improvements due to Salberger, Walsh, and Castryck-Cluckers-Dittmann-Nguyen. The main difficulty is to count integral points on lines on an affine surface in \mathbbA³, for which we develop point-counting results for curves in ℙ¹× ℙ¹. We also formulate and prove analogous results over global fields, following work by Paredes-Sasyk.

Cited by

• Counting rational points on affine hypersurfaces
• Serre's question on thin sets in projective space
• A geometric determinant method and geometric dimension growth

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