Small solutions of cubic congruences

Abstract:

Let $C({\mathbf {x}})$ be a cubic form in $n$ variables over ${\mathbf {Z}}$ and $p$ be a prime. Then for $0 < \sigma < \frac {2}{3}$ the congruence $C({\mathbf {x}}) \equiv 0(\bmod p)$ has a nonzero solution $x$ with $\max \left | {{x_i}} \right | \ll {p^{1/3 + \sigma }}$, provided that $n > 8/\sigma$, (where the constant in the $\ll$ depends on $n$ and $\sigma$).