# Question regarding a result about a homogeneous cubic equation $C(x_1, …, x_n) = 0$ with integral coefficients

I was reading "Analytic Methods for Diophantine equations and diophantine inequalities" by Harold Davenport and I came across the result (in Chapter 13) that a homogeneous cubic equation $$C(x_1, ..., x_n) = 0,$$ with integral coefficients, is soluble in integers $x_1, ..., x_n$ (not all $0$) if $n \geq 16.$

I got curious when I read this and I have two questions regarding it.

1. Has this result been improved to a smaller $n$ than $16$? I would appreciate a reference as I would be interested in reading the paper.

2. I was wondering if anyone could tell me if there is an analogous result in $\mathbb{F}_q[t]$, a polynomial ring over a finite field. I just thought if there is an analogous result maybe $n$ will be smaller in this case using some algebraic geometry, and I was curious to find out.

Thanks!

• Do you have access to Math Reviews? You could look up Davenport's paper to see whether any newer papers refer to it. – Gerry Myerson Apr 13 '14 at 12:34