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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
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