Suppose I have $n$ quadratic homogeneous polynomials $f_1, \cdots, f_n$ in $n+1$ variables over the field $\mathbb{C}$. Bezout's theorem says that generically there will be $2^n$ common zeroes.
I was wondering if/when we could say something more about these zeroes. For example, is there some criteria we can give such that we can guarantee that at least one of the zeroes is actually in $\mathbb{R}^{n+1}$ (or say $\mathbb{Q}^{n+1}$ or $\mathbb{Z}^{n+1}$)?