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.
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.
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!