2023/07/18 by Bernert, Christian, Hochfilzer, Leonhard
#11D25 #11D72 #11P55 #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2307.10294
We show that every cubic form with coefficients in an imaginary quadratic number field K/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 ≥ 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.