# Budan-Fourier theorem

2010 Mathematics Subject Classification: Primary: 12Y Secondary: 65T [MSN][ZBL]

The number of roots of an algebraic equation $f(x)=0$ comprised in an interval $(a,b), a<b$, is equal to or is smaller, by an even number, than $\tau=t_1-t_2$, where $t_1$ is the number of changes in sign in the series of derivatives of the polynomial $f(x)$ at the point $a$, i.e. in the series

$$f(a),f'(a),\ldots,f^{(n)}(a),$$

while $t_2$ is the number of changes in sign in this series at the point $b$. Each multiple root is counted according to its multiplicity. Established by F. Budan (1807) [Bu] and J. Fourier (1820) [Fo]. See also the Wikipedia article Budan's theorem.

As has already been noted by Serret [Se, p. 267], the above statement is due to Fourier [Fo]. The statement of Budan's theorem can be found in the Wikipedia article mentioned above. On this topic, see also [Ak1] and [Ak2].

An application of the Budan–Fourier theorem in numerical analysis may be found in [BoSc], where it is used in the interpolation by spline functions.

An application of the statement of (only) Budan's theorem in computer algebra may be found in [Ak3], where it is used as a no roots test.

How to Cite This Entry:
Budan-Fourier theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Budan-Fourier_theorem&oldid=25259
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article