Given a polynomial equation in multiple variables (over $\mathbb{R}$), what tools are there to estimate the set of solutions? The equation I have is of degree 2 and I want to prove that a certain set of $\mathbb{R}^4$ does not contain solutions to the equation. That set is no axis-oriented cuboid, sadly, even complicating the problem.
I'd like to know proof-methods. Numerically others have already computed that there are no solutions in the given set, but we don't have proof that it really is the case.
I'm not well acquainted with algebraic geometry and expect, that this problem might have been discussed there already, but I could not find a fitting reference. Is there even hope for general methods to exist in this situation?
Edit: The forbidden set is given by some (polynomial) inequalities, which are simple in the sense that they also have at most degree 2 and at most 2 variables occur in each.