# 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 [1], 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.

#### References

 [1] J.Ch. Sturm, Bull. de Férussac , 11 (1829) [2] A.G. Kurosh, "Higher algebra" , MIR (1972) (Translated from Russian)