This root-counting theorem was produced by the French mathematician Jacques Sturm in 1829.

In particular, we can count the total number of distinct real roots by looking at the limits as $a\rightarrow-\infty$ and $b\rightarrow+\infty$. The total number of distinct real roots will depend only on the leading terms of the Sturm sequence polynomials.

Note that deg $P_{n}<$ deg $P_{{n-1}}$, and so the longest possible Sturm sequence has deg $P+1$ terms.

Also, note that this sequence is very closely related to the sequence of remainders generated by the Euclidean Algorithm; in fact, the term $P_{i}$ is the exact same except with a sign changed when $i\equiv 2$ or $3\;\;(\mathop{{\rm mod}}4)$. Thus, the Half-GCD Algorithm may be used to compute this sequence. Be aware that some computer algebra systems may normalize remainders from the Euclidean Algorithm which messes up the sign.

For a proof, see Wolpert, N., “ Proof of Sturm’s Theorem”