I have the following polynomial:
It came up in a larger proof, and I would need in order to complete the proof to prove the following result:
Does there exist $(x,y,z,r)\in\mathbb Q^4$ such that $x\ne 0$ and
We can reformulate the problem in the following way:
Does the algebraic variety defined by
have a rational point with $X\ne 0$?
I have no idea how to tackle this problem, I have looked up several articles, but nothing seems to apply to this particular question.
Any hints or references would be greatly appreciated.