# Sturm's Theorem

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## Sturm's theorem

[′stərmz ‚thir·əm]*p*(

*x*) which lie between two given values of

*x*; the Sturm sequence of

*p*(

*x*) provides the necessary information.

## Sturm’s Theorem

a theorem that provides a basis for finding nonoverlapping intervals such that each contains one real root of a given algebraic polynomial with real coefficients. The theorem was given in 1829 by J. C. F. Sturm.

For any polynomial *f*(*x*) without multiple roots, there exists a system of polynomials

*f*(*x*) = *f*_{0}(*x*), *f*_{1}(*x*),...,*f*_{s}(*x*)

for which the following conditions are fulfilled: (1) *f*_{k}(*x*) and *f*_{k+1}(*x*), *k* = 0, 1,..., *s* – 1, do not have common roots; (2) the polynomial *f*_{s} (*x*) has no real roots; (3) it follows from *f*_{k} (α) = 0, 1 ≤ *k* ≤ *s* –1 that *f*_{k–1} (α)*f*_{k+1} (α) < 0; and (4) it follows from *f*(α) = 0 that the product *f*(*x*)*f*_{1}(*x*) is increasing at the point α. Let *w*(*c*) be the number of changes of sign in the system

*f*(*c*), *f*_{1}(*c*),...,*f*_{s}(*c*)

If the real numbers *a* and *b* (*a* < *b*) are not roots of the polynomial *f*(*x*), then the difference *w*(*a*) – *w*(*b*) is nonnegative and equal to the number of real roots of the polynomial *f*(*x*) that lie between *a* and *b*. Thus, the number line may be divided into intervals each of which contains one real root of the polynomial *f*(*x*).