Sturm theorem

If

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

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)