Small solutions of cubic congruences

Cochrane, Todd

doi:10.1090/S0002-9939-1989-0964454-X

Small solutions of cubic congruences
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by Todd Cochrane PDF

Proc. Amer. Math. Soc. 106 (1989), 333-334 Request permission

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 $\sigma$).

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