Instead of looking for roots of a polynomial $p:\Bbb R^n$, we might be interested about the points at which $p$ is "furthest" from having a root (at least locally):
What is the maximum number of isolated local maxima of the function $|p(x)|$, where $p:\Bbb R^n \to \Bbb R$ is a quadratic polynomial?
Related questions:
Proposition. The maximum number of isolated local maxima that $|p(x)|$ for any quadratic polynomial $p:\Bbb R^n \to \Bbb R$ is 1.
Proof. Clearly, $|p(x)|$ can have at least one isolated maximum as can be seen from the example $p(x) = 1 - \sum_{i=1}^n x_i^2$.
Reversely, $|p(x)|$ can not have more than one isolated maxima of. Indeed, for contradiction suppose that $|p(x)|$ had isolated local maxima at two distinct points $u,v\in \Bbb R^n$, and consider the function $f(t) = p((1-t)a+tb)$. Since $p$ is a quadratic polynomial, $f$ is a quadratic polynomial in single variable, and what is more it needs to have isolated local minima or maxima at each of the points $t=0$ and $t=1$, as $|f(t)|$ has isolated local maxima at those points. However, a quadratic function in one variable might have at most one extremum (otherwise it would have to be constant). Contradiction.
