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?



$n-1$ quadratic forms for $n$ variables
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"](http://mate.dm.uba.ar/~alidick/papers/chapter1cd.pdf) by Cattani and Dickenstein.


