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I am trying to decide whether following diophantine equation over $\mathbb{Q}[x]$ are solvable:

(a) $p(x^2 − 1) + q(x^2 + 2x + 1) + r(x^2 − 2x + 1) = x^2 + 1$

(b) $p(x^3 − 1) + q(x^4 − 1) = x^2 − 1$

For either, I get for example $\gcd(x^2-1,x^2+2x+1,x^2-2x+1) = 1$ and $\gcd(x^3-1,x^4-1) = x-1$.

So $1 \mid (x^2+1)$ and $(x-1)\mid(x^2-1)$. Does that imply that the equations are solvable? How do I determine the solutions?

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  • The equations are solvable for $gcd$ reasons. Note that a solution to the first is available by inspection. Take $p=0$, $q=r=\frac{1}{2}$. – André Nicolas May 01 '16 at 21:57

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