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

Definition 1   Let be a real polynomial in , and define the Sturm sequence of polynomials by

Here denotes the remainder of the polynomial upon division by the polynomial . The sequence terminates once one of the is zero.

Definition 2   For any number , let denote the number of sign changes in the sequence .

Theorem 1   For real numbers and that are both not roots of ,

In particular, we can count the total number of distinct real roots by looking at the limits as and . The total number of distinct real roots will depend only on the leading terms of the Sturm sequence polynomials.

Note that deg deg , and so the longest possible Sturm sequence has deg terms.

Also, note that this sequence is very closely related to the sequence of remainders generated by the Euclidean Algorithm; in fact, the term is the exact same except with a sign changed when or . 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”