Editing Sturm's theorem (section)

==Application==
Generalized Sturm sequences allow counting the roots of a polynomial where another polynomial is positive (or negative), without computing these root explicitly. If one knows an isolating interval for a root of the first polynomial, this allows also finding the sign of the second polynomial at this particular root of the first polynomial, without computing a better approximation of the root.

Let {{math|''P''(''x'')}} and {{math|''Q''(''x'')}} be two polynomials with real coefficients such that {{mvar|P}} and {{mvar|Q}} have no common root and {{mvar|P}} has no multiple roots. In other words, {{mvar|P}} and {{math|''P'{{space|hair}}Q''}} are [[coprime|coprime polynomials]]. This restriction does not really affect the generality of what follows as [[polynomial greatest common divisor|GCD]] computations allows reducing the general case to this case, and the cost of the computation of a Sturm sequence is the same as that of a GCD. 

Let {{math|''W''(''a'')}} denote the number of sign variations at {{mvar|a}} of a generalized Sturm sequence starting from {{mvar|P}} and {{math|''P'{{space|hair}}Q''}}. If {{math|''a'' < ''b''}} are two real numbers, then {{math|''W''(''a'') – ''W''(''b'')}} is the number of roots of {{mvar|P}} in the interval <math>(a,b]</math> such that {{math|''Q''(''a'') > 0}} minus the number of roots in the same interval such that {{math|''Q''(''a'') < 0}}. Combined with the total number of roots of {{mvar|P}} in the same interval given by Sturm's theorem, this gives the number of roots of {{mvar|P}} such that {{math|''Q''(''a'') > 0}} and the number of roots of {{mvar|P}} such that {{math|''Q''(''a'') < 0}}.<ref name="bpr" />