An equation with polynomial unknowns

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Crankornotcrank
Posts: 7
Joined: Mon Nov 09, 2015 4:37 pm UTC

An equation with polynomial unknowns

Postby Crankornotcrank » Mon Nov 09, 2015 4:49 pm UTC

We have the identity (X + Y) (- X + Y)^3 = X (- X - 2Y)^3 + Y (2X + Y)^3.
Thus, there are three different polynomials f(X, Y), g(X, Y) and h(X, Y), with rational coefficients, such that
(X + Y) f(X)^3 = X g(X)^3 + Y h(X)^3. (Take f(X) = (- X + Y, g(X) = - X - 2Y and h(X) = 2X + Y.)
Do you know if there exist three different polynomials F(X, Y), G(X, Y) and H(X, Y), with rational coefficients, such that
(X + Y) F(X, Y)^5 = X G(X, Y)^5 + Y H(X, Y)^5.
(I think that the answer is no, but I can't prove it.)
Thanks in advance.

Crankornotcrank
Posts: 7
Joined: Mon Nov 09, 2015 4:37 pm UTC

Re: An equation with polynomial unknowns

Postby Crankornotcrank » Wed Nov 11, 2015 7:19 am UTC

I think that I have found an easy proof. If anybody is interested, I can give it.
C.


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