Example of a rational function such that : $(f(x))^{3} + (g(x))^{3} + (h(x))^{3}=x$

Solution 1:

$(a^4-2a)^3+(a^3+1)^3+(2a^3-1)^3=(a^4+a)^3$

There are more at http://sites.google.com/site/tpiezas/010 Section 2.

This is Number Theory, not real analysis.

Solution 2:

You may be interested to know identities are known that can bring this higher:

$a_1(x)^3+a_2(x)^3+a_3(x)^3 = x$

$a_1(x)^5+a_2(x)^5+a_3(x)^5+...+a_6(x)^5 = x$

$a_1(x)^7+a_2(x)^7+a_3(x)^7+a_4(x)^7+...+a_8(x)^7 = x$

where the given number of addends $a_i$ are rational functions in terms of x, hence can be treated as "Waring-like problems". The case k = 3 mentioned by Esteban is cited in Yuri Manin's book Cubic Forms (but has been solved earlier by Ryley), while k = 5 and 7 have been solved by Choudhry. Interestingly, k = 7 involves polynomials where the coefficients have 33 digits! One has to solve the simultaneous equations,

$a^2+m^5b^2 = c^2+m^5d^2$

$a^4+m^3b^4 = c^4+m^3d^4$

which, for m = 2, can be reduced to an elliptic curve and has an infinite number of integer solutions (after multiplying out the denominators), though the "smallest" one has 33 digits.

P.S. If anybody can solve k = 9, I'll be interested to know.

See, "Waring-Like Problems" at http://sites.google.com/site/tpiezas/001b.