If I have a system of linear equations, $A x = c$, with $A$ an $n\times n$ complex matrix, it is relatively easy to see that the set of matrices $A$ for which there is no (complex) solution has measure zero, as this is the set of matrices such that $\det(A) = 0$.
Can something similar be said for systems of quadratic equations?
More precisely, consider a system of $n$ quadratic equations in $n$ variables, which I can always write as
$$
\boldsymbol x^\dagger A_i \boldsymbol x + \boldsymbol b_i \cdot \boldsymbol x + c_i = 0, \quad i=1,…, n,
$$
where $A_i$ are $n\times n$ complex matrices, $\boldsymbol b_i\in\mathbb C^n$ and $c_i\in\mathbb C$.
Does this system have a solution for almost all values of the parameters?
In other words, if a given choice of parameters corresponds to no solutions, is it always true that an infinitesimal change of parameters will give me a system which has solutions?
Best Answer
Yes. The magic words are "elimination theory" and "resultant". In essence, the system has a solution unless some determinant (the iterated resultant) vanishes.