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Sturm’s theorem∗ rspuzio† 2013-03-21 17:38:39 This root-counting theorem was produced by the French mathematician Jacques Sturm in 1829. Definition 1. Let P (x) be a real polynomial in x, and define the Sturm sequence of polynomials P0 (x), P1 (x), . . . by P0 (x) = P (x) P1 (x) = P 0 (x) Pn (x) = −rem(Pn−2 , Pn−1 ), n ≥ 2 Here rem(Pn−2 , Pn−1 ) denotes the remainder of the polynomial Pn−2 upon division by the polynomial Pn−1 . The sequence terminates once one of the Pi is zero. Definition 2. For any number t, let varP (t) denote the number of sign changes in the sequence P0 (t), P1 (t), . . .. Theorem 1. For real numbers a and b that are both not roots of P (x), #{distinct real roots of P in (a, b)} = varP (a) − varP (b) In particular, we can count the total number of distinct real roots by looking at the limits as a → −∞ and b → +∞. The total number of distinct real roots will depend only on the leading terms of the Sturm sequence polynomials. Note that deg Pn < deg Pn−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 Pi is the exact same except with a sign changed when i ≡ 2 or 3 (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” ∗ hSturmsTheoremi created: h2013-03-21i by: hrspuzioi version: h36013i Privacy setting: h1i hTheoremi h11A05i h26A06i † This text is available under the Creative Commons Attribution/Share-Alike License 3.0. You can reuse this document or portions thereof only if you do so under terms that are compatible with the CC-BY-SA license. 1