Intersection of quadrics

Question

Having two quadratic surfaces in the form of implicit equations, what is the simplest way to decide whether they intersect or not? Finding the intersection itself is not necessary.

Answer 1

This question is not totally precise, but I'll try to piece together an answer. First, you should decide whether or not your quadrics are projective. Also, you should be asking which field you are working over. For example $$ \textbf{Proj}\left( \frac{\mathbb{R}[x,y,z,w]}{x^2 + y^2 + z^2 + w^2} \right) $$ has no solutions, hence it's just the empty variety which cannot intersect any other quadrics in $\mathbb{P}^3(\mathbb{R})$. If you look in the characteristic $p$ setting, you may be able to construct a pair of quadrics which do not intersect (this could be an interesting problem to construct a pair).

If you are working over $\mathbb{C}$, you should expect an intersection. Consider the case of two quadric surfaces intersecting in $\mathbb{P}^3(\mathbb{C})$, presented as $$ \textbf{Proj}\left( \frac{\mathbb{C}[x,y,z,w]}{(f(x,y,z,w),g(x,y,z,w))} \right) $$ if we intersect with a hyperplane, for example $w=0$, we get the intersection of conics $$ \textbf{Proj}\left( \frac{\mathbb{C}[x,y,z]}{(f(x,y,z,0),g(x,y,z,0))} \right) $$ which intersects at four points (including multiplicity) by Bezout's theorem. So in this case you are forced to have intersections of your quadrics.