Cubic Forms in 14 Variables

The result can be rephrased in geometric language to say that any projective cubic hypersurface defined over Q, of dimension at least 12, has a Q-point. Davenport’s result was extended to arbitrary number fields by Pleasants [9], and it would be interesting to know whether Theorem 1 could similarly be extended. These results can be seen as an attempt to extend the classical theorem of Meyer (1884) from quadratic forms to cubic forms. Meyer showed that any indefinite quadratic from over Z in 5 or more variables must represent zero. Indeed Meyer’s result was generalized by Minkowski , who showed that a quadratic form over Z, in any number of variables, represents zero if and only if it represents zero over every completion of Q. It is a well-known fact that this local condition is automatically satisfied for Qp as soon as n ≥ 5. Thus Meyer’s result requires only the condition of indefiniteness. The analogous fact for cubic forms is that p-adic zeros exist whenever n ≥ 10, see Davenport [6, Chapter 18] for example. Of course the condition for R holds for any n in this case. Thus it is natural to conjecture that Theorem 1 should hold as soon as n ≥ 10. Indeed for smaller values of n one might expect that it should suffice for the form C(x) to represent zero in each field Qp. However when n = 3 or 4 it is possible for a cubic form to have zeros in every completion of Q, without there being a global zero. This is shown by the examples