Cubic forms in thirty-two variables

Abstract
It is proved that if C(xu...,*„) is any cubic form in n variables, with integral coefficients, then the equation C{xu ...,*„) = 0 has a solution in integers xXi...,xn, not all 0, provided n is at least 32. The proof is based on the Hardy-Littlewood method, involving the dissection into parts of a definite integral, but new principles are needed for estimating an exponential sum containing a general cubic form. The estimates obtained here are conditional on the form not splitting in a particular manner; when it does so split, the same treatment is applied to the new form, and ultimately the proof is made to depend on known results.

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