bounds for circle method While browsing through Davenport's lecture notes "Analytic methods for diophantine equations and Diophantine inequalities", near the end of chap 8 I came across the statement that for the equation
$c_1 x_1^k + ... + c_s x_s^k = 0$,
if the $c_i$ are non-zero and, if $k$ is even, the sign of $c_i$ are not all the same, then  this equation has infinitely many integer solutions if $s\ge k^2+1$.  Davenport then noted that the bound $k^2+1$ is optimal if $k+1$ is prime, and that "for most values of $k$ smaller value than $k^2+1$ will suffice.
Question: What's the current record for $s$ for small $k = 3, 5, 7, 8, ...$ " ?  What about conjectural/expected "optimal" value for $s$ for small $k$?
THANKS!
 A: In general, what the circle method gives you is an asymptotic formula for the number of solutions to the equation, which will be of the form $C P^{s-k}$ for some non-negative $C$. We are currently (with the help of recent advances on Vinogradov's mean value theorem) able to establish such asymptotic formulae when $s>k^2 + O(k)$ (see Wooley 2012 IMRN).
If one doesn't require an asymptotic formula and is instead happy with infinitely many solutions, it is possible to restrict the base set from the integers to smooth numbers (i.e. no large prime divisors). One can then establish a similar asymptotic formula for the number of solutions in as few as $s>k \log k +O(k \log \log k)$ variables.
Generally, the expectation is that such an asymptotic formula should hold whenever $s>2k$, and possibly (with a diverging factor $C$) when $s>k$. However, as things stand this is very much pie in the sky territory!
The elephant in the room, however, is the constant $C$. In Davenport's book you can see that $C$ has a representation as the product over the densities of solutions to your equation over all local fields $\mathbb R$ and $\mathbb Q_p$, and this is where the real bottleneck sits. The general rule of thumb is that you can expect a value that is smaller than $k^2+1$ whenever you can prove that there are no $p$-adic obstructions.
Understanding the details and particulars of this last question is in general still open, but Mike Knapp and Hemar Godinho have published papers on this, which you could check out.
