Given a polynomial with roots $x_1,\ldots,x_n$ and real coefficients, it can be written $$ P(x)=\prod_{i=1}^n \left( x-x_i \right);$$ define Newton polynomials $$s_k(x_1,\ldots,x_n):=\sum_{i=1}^n x_i^k$$ and the inverse metric matrix $g^{-1}$ (of some pseudo-Riemannian manifold) $$ g^{kl} := kls_{k+l-2}.$$
Theorem[Sturm]: the roots of $P(x)=0$ are real and distinct iff $g$ is positive definite.
Given the above, how one can show the following?
Exercise: if $P(x)=0$ has distinct roots, then it has precisely $m$ real roots with $(m+k,k)$ being the signature of matrix $g$ and $m+2k=n,$ where $n$ is the dimension of the manifold, i.e. the rank of the metric matrix $g.$