The circle method and diagonal cubic forms - Heath-Brown, D (1998)
We use the Hardy-Littlewood circle method, in the form developed by Heath-Brown in 1996, to investigate the number of integer zeros of diagonal cubic forms. The results are subject to unproved hypotheses concerning certain Hasse-Weil L-functions. For six variables we show that there are O(P3+ε) zeros up to height P, for any ε > 0. For four variables we show that there are O(P3/2+ε) such zeros, excluding any that lie on rational lines in the corresponding surface.