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

TL;DR

This paper extends classical results on cubic forms to imaginary quadratic fields, proving non-trivial zeros in at least 14 variables and establishing conditions for rational solutions and lines in cubic hypersurfaces.

Contribution

It generalizes Heath-Brown's results from rational to imaginary quadratic fields and provides new bounds for rational solutions and lines in cubic forms.

Findings

Cubic forms over imaginary quadratic fields in ≥14 variables represent zero non-trivially.

Pairs of rational cubic forms in ≥627 variables have non-trivial rational solutions.

Every rational cubic hypersurface in ≥33 variables contains a rational line.

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.