Integral points on cubic hypersurfaces
Abstract
Let g be a cubic polynomial with integer coefficients and n>9 variables, and assume that the congruence g=0 modulo pk is soluble for all prime powers pk. 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.
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