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!

lower boundof the number integer solutions say for k=3? $\endgroup$