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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!

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  • $\begingroup$ Do you have access to Math Reviews? You could look up Davenport's paper to see whether any newer papers refer to it. $\endgroup$ Apr 13, 2014 at 12:34

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Heath-Brown, Cubic forms in 14 variables, Invent. Math 170 (2007) 199-230, improves Davenport's result from 16 to 14. It appears the natural conjecture is that 10 will do.

I didn't find any improvements on Heath-Brown's result at Math Reviews.

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  • $\begingroup$ I see. Thank you very much! $\endgroup$
    – Tom Mosher
    Apr 14, 2014 at 18:22

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