Sufficient condition for pair of real quadrics to have real intersection

Question

In the following, when I talk about the zero of a homogeneous polynomial I always mean a projective zero.

Let $ q $ be a real quadric. Then $ q $ has a real zero if and only if $ q $ has indefinite signature as a quadratic form (in other words, isotropic).

Let $ q_1 $ and $ q_2 $ be real quadrics. What is a sufficient condition for $ q_1,q_2 $ to have a real intersection? (in other words a shared real zero).

Note that while it is necessary for $ q_1 $ and $ q_2 $ to both be of indefinite signature that is not sufficient. For example $$ q_1(x_1,x_2,x_3)=x_1^2+3x_1x_2+x_1x_3+x_2x_3-x_3^2 $$ and $$ q_2(x_1,x_2,x_3)=-x_1x_2+x_2^2-x_1x_3-x_2x_3+x_3^2 $$ both have signature $ (2,1,0) $, but they have no real intersection. To see why, note that $ q_1+q_2=x_1^2+2x_1x_2+x_2^2=(x_1+x_2)^2 $. So at any shared 0 we have $ (x_1+x_2)^2=0 $. Thus at any shared real zero we have $ x_2=-x_1 $. Plugging this into $ q_1 $ we have that at any shared real zero $ -2x_1^2-x_3^2=0 $. So we must have $ x_1=0=x_3 $ which in turn implies that at any shared real zero $ x_2=-x_1=0 $. Thus $ q_1,q_2 $ have no real intersection.

Note on the complex case: Bezout's theorem says that if $ q_1,q_2 $ are quadrics in 3 or more variables then that is sufficient to guarantee a shared complex zero.

Answer 1

Theorem: The two quadratic forms $q_1, q_2$ have a common nontrivial zero defined over $\mathbb{R}$ if and only if $a q_1 + b q_2$ is not definite for every pair $a,b \in \mathbb{R}$.

Note that "not definite" means either indefinite or semi-definite.

This is an old theorem with a very large number of proofs. I published a survey paper with Nandita Sahajpal that contains a proof of this theorem. The reference is

"Pairs of Quadratic Forms over the Real Numbers", Bulletin of the Irish Mathematical Society, Volume 91, Summer 2023, pages 49-72.

The paper is a self-contained presentation starting with only the basics of quadratic form theory.

This paper contains many other topics on pairs. The theorem in question is proved in Section 4. The paper contains references (reference [8] in particular) that give lots of other references to hunt down many proofs of this theorem.