Integral points on cubic surfaces: heuristics and numerics

TL;DR

This paper introduces a heuristic to estimate the density of integer points on affine cubic surfaces, comparing it with existing predictions and testing it against numerical data for various surface families.

Contribution

It develops a new heuristic for integer point density on cubic surfaces and adapts it for both smooth and singular cases, connecting with existing theories and numerical data.

Findings

Heuristic aligns with Heath-Brown's predictions for sums of three cubes.

Matches asymptotic formulas related to Zagier's work on the Markoff surface.

Successfully tested against numerical data for multiple cubic surface families.

Abstract

We develop a heuristic for the density of integer points on affine cubic surfaces. Our heuristic applies to smooth surfaces defined by cubic polynomials that are log K3, but it can also be adjusted to handle singular cubic surfaces. We compare our heuristic to Heath-Brown's prediction for sums of three cubes, as well as to asymptotic formulae in the literature around Zagier's work on the Markoff cubic surface, and work of Baragar and Umeda on further surfaces of Markoff-type. We also test our heuristic against numerical data for several families of cubic surfaces.