Let g be a cubic polynomial with integer coefficients and n>9 variables, and
assume that the congruence g=0 modulo p^k is soluble for all prime powers p^k.
We show that the equation g=0 has infinitely many integer solutions when the
cubic part of g defines a projective hypersurface with singular locus of
dimension <n-10. The proof is based on the Hardy-Littlewood circle method.Comment: 18 page
