2023/07/18 by Brandes, Julia, Dietmann, Rainer, Leep, David B.
#14G05 #14J70 #FOS: Mathematics #Number Theory (math.NT) #Primary: 11D72. Secondary: 11E76
paper · doi:10.48550/arxiv.2307.09449
We show that any rational cubic hypersurface of dimension at least 33 defined over a number field K vanishes on a K-rational projective line, reducing the previous lower bound of Wooley by two. For K=\mathbb Q we can reduce the bound to 29. The main ingredients are a result on linear spaces on quadratic forms over suitable non-real quadratic field extensions, and recent work of Bernert and Hochfilzer on cubic forms over imaginary quadratic number fields for the rational case.