# Sturm theorem

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If (*)

is a Sturm series on the interval , , and is the number of variations of sign in the series (*) at a point (vanishing terms are not taken into consideration), then the number of distinct roots of the function on the interval is equal to the difference .

A Sturm series (or Sturm sequence) is a sequence of real-valued continuous functions (*) on having a finite number of roots on this interval, and such that

1) ;

2) on ;

3) from for some  and given in it follows that ;

4) from for a given  it follows that for sufficiently small ,  This theorem was proved by J.Ch. Sturm , who also proposed the following method of constructing a Sturm series for a polynomial with real coefficients and without multiple roots: , , and, if the polynomials are already constructed, then as one should take minus the remainder occurring in the process of dividing by . Here, will be a non-zero constant.

How to Cite This Entry:
Sturm theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Sturm_theorem&oldid=17606
This article was adapted from an original article by I.V. Proskuryakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article