Sturm theorem

If

$$ \tag{* } f _ {0} ( x) \dots f _ {s} ( x) $$

is a Sturm series on the interval $ [ a, b] $, $ a < b $, and $ w( x) $ is the number of variations of sign in the series (*) at a point $ x \in [ a, b] $ (vanishing terms are not taken into consideration), then the number of distinct roots of the function $ f _ {0} $ on the interval $ [ a, b] $ is equal to the difference $ w( a)- w( b) $.

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

1) $ f _ {0} ( a) f _ {0} ( b) \neq 0 $;

2) $ f _ {s} ( x) \neq 0 $ on $ [ a, b] $;

3) from $ f _ {k} ( c) = 0 $ for some $ k $ $ ( 0 < k < s) $ and given $ c $ in $ [ a, b] $ it follows that $ f _ {k-1} ( c) f _ {k+1} ( c) < 0 $;

4) from $ f _ {0} ( c) = 0 $ for a given $ c $ $ ( a < c < b) $ it follows that for sufficiently small $ \epsilon > 0 $,

$$ f _ {0} ( x) f _ {1} ( c) < 0 \ ( c- \epsilon < x < c); $$

$$ f _ {0} ( x) f _ {1} ( c) > 0 \ ( c < x < c + \epsilon ). $$

This theorem was proved by J.Ch. Sturm [1], who also proposed the following method of constructing a Sturm series for a polynomial $ f ( x) $ with real coefficients and without multiple roots: $ f _ {0} ( x) = f ( x) $, $ f _ {1} ( x) = f ^ { \prime } ( x) $, and, if the polynomials $ f _ {0} ( x) \dots f _ {k} ( x) $ are already constructed, then as $ f _ {k+1} ( x) $ one should take minus the remainder occurring in the process of dividing $ f _ {k-1} ( x) $ by $ f _ {k} ( x) $. Here, $ f _ {s} ( x) $ 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)

Comments

The coefficients of the polynomials in the Sturm series must belong to a real-closed field. The algorithm to determine a Sturm series for a polynomial $ f _ {0} ( x) $ can be described as follows:

$$ f _ {0} ( x) = f ( x),\ f _ {1} ( x) = f ^ { \prime } ( x) , $$

$$ f _ {0} ( x) = q _ {1} ( x) f _ {1} ( x) - f _ {2} ( x) ,\ \mathop{\rm deg} f _ {2} ( x) < \mathop{\rm deg} f _ {1} ( x) , $$

$$ \dots \dots \dots \dots $$

$$ f _ {k-} 1 ( x) = q _ {k} ( x) f _ {k} ( x) - f _ {k+1} ( x) ,\ \mathop{\rm deg} f _ {k+1} ( x) < \mathop{\rm deg} f _ {k} ( x), $$

$$ \dots \dots \dots \dots $$

$$ f _ {s-1} ( x) = q _ {s} ( x) f _ {s} ( x) , $$

so $ f _ {s} ( x) $ is a non-zero constant.

References

[a1]	N. Jacobson, "Basic algebra" , I , Freeman (1974)
[a2]	L.E.J. Dickson, "New first course in the theory of equations" , Wiley (1939)
[a3]	B.L. van der Waerden, "Algebra" , 1 , Springer (1967) (Translated from German)