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arXiv · 2307.10294

Cubic forms over imaginary quadratic number fields and pairs of rational cubic forms

Christian Bernert·Leonhard Hochfilzer
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Abstract

We show that every cubic form with coefficients in an imaginary quadratic number field $K/\mathbb{Q}$ in at least $14$ variables represents zero non-trivially. This builds on the corresponding seminal result by Heath-Brown for rational cubic forms. As an application we deduce that a pair of rational cubic forms has a non-trivial rational solution provided that $s \geq 627$. Furthermore, we show that every rational cubic hypersurface in at least $33$ variables contains a rational line, and that every rational cubic form in at least $33$ variables has "almost-prime" solutions.

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Christian Bernert, Leonhard Hochfilzer. 2023-07-18. Cubic forms over imaginary quadratic number fields and pairs of rational cubic forms. https://arxiv.org/abs/2307.10294

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