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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.

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    $\begingroup$ It is known that every positive integer $> 13792$ is the sum of sixteen fourth powers of integers, and so $16$ rational fourth powers suffice. I suspect one can do substantially better with some positive and some negative signs. In particular, Ramanujan found an infinite collection of representations of $k = 81$ in the form $x^4 + y_{1}^{4} + y_{2}^{4} - y_{3}^{4} - y_{4}^{4}$. $\endgroup$
    – Jeremy Rouse
    Commented Oct 25, 2016 at 12:58
  • $\begingroup$ The title refers to rational $k$, the body, positive integer $k$. $\endgroup$
    – Gerry Myerson
    Commented Oct 25, 2016 at 22:22
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    $\begingroup$ According to the review by Greaves, the paper L Habsieger, Représentations des groupes et identités polynomiales, Sém. Théor. Nombres Bordeaux (2) 3 (1991), no. 1, 1–11, contains "a new result on the 'rational'' Waring's problem for fourth powers." It is possible that there is more in the author's Applications of group representation theory to the easier Waring problem, J. Number Theory 45 (1993), no. 1, 92–111. $\endgroup$
    – Gerry Myerson
    Commented Oct 25, 2016 at 22:27

2 Answers 2

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(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.

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This problem (for integers) is known as "An Easier Waring's Problem". In the article (1934) with the same name Wright proved that for degree $4$ one need $v(4)$ summands where $8\le v(4)\le 12$. This upper bound is not trivial because using simple identity $$(x+4)^4-2(x+3)^4+2(x+1)^4-x^4=48x+96$$ you can only prove that $v(4)\le 14$.

In 1941 Hunter proved (see The Representation of Numbers By Sums of Fourth Powers) that $v(4)\in\{9,10\}.$

Probably this result was not improved.

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  • $\begingroup$ Is upper bound 10 also based on polynomial identities? $\endgroup$
    – Fedor Petrov
    Commented Oct 25, 2016 at 15:47
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    $\begingroup$ @Fedor Petrov Yes, he used $4$ identities with $7$, $7$, $10$, $10$ summands respectively. $\endgroup$
    – Alexey Ustinov
    Commented Oct 26, 2016 at 5:34

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