Editing Sturm's theorem (section)

==The theorem==
The '''Sturm chain''' or '''Sturm sequence''' of a [[univariate polynomial]] {{math|''P''(''x'')}} with real coefficients is the sequence of polynomials <math>P_0, P_1, \ldots,</math> such that 
:<math>\begin{align}
P_0&=P,\\
P_1&=P',\\
P_{i+1}&=-\operatorname{rem}(P_{i-1},P_i),
\end{align}</math>
for {{math|''i'' ≥ 1}}, where {{math|''P'{{void}}''}} is the [[derivative]] of {{mvar|P}}, and <math>\operatorname{rem}(P_{i-1},P_i)</math> is the remainder of the [[Euclidean division of polynomials|Euclidean division]] of <math>P_{i-1}</math> by <math>P_{i}.</math> The length of the Sturm sequence is at most the degree of {{mvar|P}}.

The number of [[sign variation]]s at {{mvar|ξ}} of the Sturm sequence of {{mvar|P}} is the number of sign changes (ignoring zeros) in the sequence of real numbers
:<math>P_0(\xi), P_1(\xi),P_2(\xi),\ldots.</math>
This number of sign variations is denoted here {{math|''V''(''ξ'')}}. 

Sturm's theorem states that, if {{mvar|P}} is a [[square-free polynomial]],  the number of distinct real roots of {{mvar|P}} in the [[half-open interval]] {{math|(''a'', ''b'']}} is {{math|''V''(''a'') − ''V''(''b'')}} (here, {{mvar|a}} and {{mvar|b}} are real numbers such that {{math|''a'' < ''b''}}).<ref name="bpr" />

The theorem extends to unbounded intervals by defining the sign at {{math|+∞}} of a polynomial as the sign of its [[leading coefficient]] (that is, the coefficient of the term of highest degree). At {{math|–∞}} the sign of a polynomial is the sign of its leading coefficient for a polynomial of even degree, and the opposite sign for a polynomial of odd degree.

In the case of a non-square-free polynomial, if neither {{mvar|a}} nor {{mvar|b}} is a multiple root of {{mvar|p}}, then {{math|''V''(''a'') − ''V''(''b'')}} is the number of ''distinct'' real roots of {{mvar|P}}.

The proof of the theorem is as follows: when the value of {{mvar|x}} increases from {{mvar|a}} to {{mvar|b}}, it may pass through a zero of some <math>P_i</math> ({{math|''i'' > 0}}); when this occurs, the number of sign variations of <math>(P_{i-1}, P_i, P_{i+1})</math> does not change. When {{mvar|x}} passes through a root of <math>P_0=P,</math> the number of sign variations of <math>(P_0, P_1)</math> decreases from 1 to 0. These are the only values of {{mvar|x}} where some sign may change.