Is any known method capable of bounding (either definitely or as an estimate) the maximum number of unknowns $x$, $y\_1$, .. $y\_n$
for which there is guaranteed to be a rational solution of $x^4 \pm y\_1^4 \pm y\_2^4 .. \pm y\_n^4 = k$ for any given positive integer $k$, where the sign options are independent?


The 4th power Waring's problem gives an upper estimate of 18 integer fourth powers in total (choosing all signs positive). But presumably allowing rational values and the sign choices must reduce this bound significantly.


This isn't merely a whimsical problem off the top of my head. I do have a sound reason for being interested in it.


Edit: Many thanks for the replies and comments. Although these were all equally useful and interesting in their own ways, Fedor Petrov's best suits my purpose (in relation to rational numbers). So I've marked that as the "accepted" reply.



Rational solutions of $x^4 \pm y_1^4 \pm .. y_n^4 = |k|$ for given rational $k$
(Not a complete answer.)


Rational solutions with negative signs allowed may be found by polynomial formulae, like $$x^4+(2x+17)^4-(x+16)^4-(2x+15)^4=-4080(x+8),$$
this may be an arbitrary rational number, thus 4 fourth powers are enough. 


