Sturm's Theorem for Polynomials
Let be the number of real roots of an algebraic equation with real coefficients whose real roots are simple over an interval and are not or . Then , the difference between the number of sign changes of the Sturm chain evaluated at and at .[more]
By subdividing an interval until every subinterval contains at most one root, one can locate subintervals containing all the real roots in the original interval.[less]
The Sturm chain of a polynomial is the sequence of polynomials:
Here and are the polynomial quotient and remainder of . The chain ends when the polynomial is a constant.