Sturm's theorem

Remarks

The sequence is the Euclidean algorithm for $p$ and $p'$ with sign changes, so $p_m$ is a greatest common divisor of $p$ and $p'$, a nonzero constant here. As $\xi$ increases, $V(\xi)$ can only change when $\xi$ passes a root of some $p_i$. At a root of $p_i$ with $0 < i < m$, the relation $p_{i-1} = q_i p_i - p_{i+1}$ forces $p_{i-1}$ and $p_{i+1}$ to have opposite signs, so $V$ does not change. At a root of $p$, the signs of $p$ and $p'$ go from opposite to equal, so $V$ drops by exactly one. Example: for $p = x^2 - 2$, the sequence is $x^2 - 2$, $2x$, $2$; then $V(0) = 1$ and $V(2) = 0$, so there is one root in $(0, 2)$, namely $\sqrt 2$. The theorem also holds without the hypothesis on multiple roots, still counting distinct roots, but the proof then divides the sequence by $p_m$.

Going deeper

For a student

Unlike Descartes' rule of signs, which only gives an upper bound, Sturm's theorem gives the exact number of distinct roots. By bisection, it isolates every real root in an interval as small as one likes, which is the basis of exact real root isolation in computer algebra. Taking $a$ below and $b$ above all roots, it counts all real roots of $p$, using the signs of leading coefficients only. The same argument works over any real closed field, where the intermediate value theorem holds for polynomials.