Editing Sturm's theorem (section)

==Generalization==
Sturm sequences have been generalized in two directions. To define each polynomial in the sequence, Sturm used the negative of the remainder of the [[Euclidean division of polynomials|Euclidean division]] of the two preceding ones. The theorem remains true if one replaces the negative of the remainder by its product or quotient by a positive constant or the square of a polynomial. It is also useful (see below) to consider sequences where the second polynomial is not the derivative of the first one.

A ''generalized Sturm sequence'' is a finite sequence of polynomials with real coefficients
:<math>P_0, P_1, \dots, P_m</math>
such that
* the degrees are decreasing after the first one: <math>\deg P_{i} <\deg P_{i-1}</math> for {{math|1=''i'' = 2, ..., ''m''}};
* <math>P_m</math> does not have any real root or has no sign changes near its real roots.
* if {{math|''P<sub>i</sub>''(''ξ'') {{=}} 0}} for {{math|0 < ''i'' < ''m''}} and {{mvar|ξ}} a real number, then {{math|''P''<sub>''i'' −1 </sub>(''ξ'') ''P''<sub>''i'' + 1</sub>(''ξ'') < 0}}.

The last condition implies that two consecutive polynomials do not have any common real root. In particular the original Sturm sequence is a generalized Sturm sequence, if (and only if) the polynomial has no multiple real root (otherwise the first two polynomials of its Sturm sequence have a common root).

When computing the original Sturm sequence by Euclidean division, it may happen that one encounters a polynomial that has a factor that is never negative, such a <math>x^2</math> or <math>x^2+1</math>. In this case, if one continues the computation with the polynomial replaced by its quotient by the nonnegative factor, one gets a generalized Sturm sequence, which may also be used for computing the number of real roots, since the proof of Sturm's theorem still applies (because of the third condition). This may sometimes simplify the computation, although it is generally difficult to find such nonnegative factors, except for even powers of {{mvar|x}}.