Consider $n$ variables $X = \{x_i\}_{i=1}^n$. Consider two 2nd degree equations over X:
$$P_k^2(x) = a_0^{(k)} + \sum_{i = 1}^n a_i^{(k)} x_i + \sum_{i = 1}^n\sum_{j = 1}^n a_{ij}^{(k)} x_ix_j = 0 $$ for $k \in \{ 1, 2 \}$ s.t. $a^k_{ij} \neq 0$ for some i,j.
What is the maximum no. of points of intersection of $P_1^2(x)$ and $P_2^2(x)$, assuming that no $P_k^2(x)$ is contained in some proper subspace of span($X$), where span($X$) is a vector space over field $\mathbb{R}$ of dimension $n$.
I know that if $n = 2$, then we get two conics, we have at most 4 points of intersection. Also if we don't put the condition that $P_k^2(x)$ is not contained in some proper subspace of span($X$), then we can easily see there are infinite points of intersection for the following polynomials over 4 variables: $x^2 + y^2 = 4$ and $u^2 + v^2 = 4$.
Edit. On googling, I was able to find this : http://homepages.warwick.ac.uk/~masda/3folds/qu.pdf However, I am not able to understand the math here. Can anyone just post the final results if they understand it (if it is related).