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Integral points on cubic surfaces: heuristics and numerics

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arxiv 2407.16315 v1 pith:QS3SAOO3 submitted 2024-07-23 math.NT

Tim Browning , Florian Wilsch This is my paper · ORCID
classification math.NT
keywords cubicsurfacesheuristicpointsworkadjustedaffineapplies
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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.

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  1. The Mordell-Schinzel conjecture for cubic diophantine equations

    math.NT 2024-12 conditional novelty 7.0 of 10

    The Mordell-Schinzel conjecture is proven for cubic equations of the form xyz = A(x) + B(y) - c when A and B have degree 3.

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