$n-1$ quadratic forms for $n$ variables If we have $n-1$ quadratic forms for $n$ variables $x_i$,
$$p_i(x) = M^{(i)}_{jk} x_j x_k$$
for $1\leq i \leq n-1$ and $1 \leq j,k \leq n$ then the zeros of all $p_i(x)$,
$$p_i(x) = 0$$
is generically $0$ dimensional in projective space, i.e. points. I guess generically there are $2^{n-1}$ of such points.
What can be said about this set of points generically? For instance, is there a necessary and sufficient condition on the quadratic forms $M^{(i)}$ such that indeed we have a zero dimensional set of solutions? Is it true that the set of points are roots of a $2^{n-1}$ order polynomial in one variable?
 A: To reach a satisfactory understanding of the problem at hand, I think you need to learn about multidimensional resultants (see below for where to get started).
Working over the field $\mathbb{C}$, let $F_1(x),\ldots, F_n(x)$ be $n$ homogeneous polynomials of respective degrees $d_1,\ldots,d_n$. Then there is a polynomial in the coefficients of these forms called the resultant $\mathrm{Res}(F_1,\ldots,F_n)$ which is zero iff there exists $x\in\mathbb{C}^n\backslash\{0\}$ such that $F_1(x)=0,\ldots,F_n(x)=0$.
This resultant is multihomogeneous of degree $\prod_{j\neq i}d_j$ in the coefficients of the form $F_i$.
Now for your situation, one should consider $\mathrm{Res}(p_1,\ldots,p_{n-1},u)$
where $u(x)$ is a generic linear form. If I remember correctly, the common zero set in projective space of your $n-1$ quadratics is zero dimensional iff the above polynomial in the coefficients of $u$ does not vanish identically (with the $p$'s fixed).
Finally, if this zero dimensional condition is satisfied and you look for the coordinates of the solutions of the system along a fixed axis, these indeed are the roots of a degree $2^{n-1}$ polynomial in one variable. This follows from the above polynomial being of degree $2^{n-1}$ with respect to $u$.
A great introductory reference for multidimensional resultants is the book chapter "Introduction to residues and resultants" by Cattani and Dickenstein.
A: It turns out this paper has a lot of useful information on resultants (more than enough to answer the original question): Morozov and Shakirov - Analogue of the identity Log Det = Trace Log for resultants.
