Budan-Fourier theorem

2020 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

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

Comments

As has already been noted by Serret[Se], footnote 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.

References

[Ak1]	Akritas, Alkiviadis G.,"On the Budan–Fourier Controversy", ACM-SIGSAM Bulletin,15, No. 1 (1981) pp. 8–10 [1]
[Ak2]	Akritas, Alkiviadis G., "Reflections on a Pair of Theorems by Budan and Fourier", Mathematics Magazine, 55, No.5 (1982) pp. 292–298 [2]
[Ak3]	Akritas, Alkiviadis G., "Vincent's Theorem of 1836: Overview and Future Research", Journal of Mathematical Sciences, 168, (2010) pp. 309–325 [3]
[AlMaCh]	P.S. Alexandroff, A.T. Markuschewitsch, A.J. Chintschin, "Encyclopaedia of elementary mathematics", 2. Algebra, Moscow-Leningrad (1951) pp. 331 (In Russian) MR0080060 Zbl 0365.00003
[BoSc]	C. de Boor, I.J. Schoenberg, "Cardinal interpolation and spline functions VIII. The Budan–Fourier theorem for splines and applications." K. Bohmer (ed.) G. Meinardus (ed.) W. Schempp (ed.), Spline functions, Lect. notes in math., 501, Springer (1976) MR0493059 Zbl 0319.41010
[Bu]	Budan, F. D., " Nouvelle méthode pour la résolution des équations numériques", Paris: Courcier (1807) google books
[Fo]	J.P.J. Fourier, "Sur l'usage du théoréme de Descartes dans la recherche des limites des racines", Bulletin des Sciences, par la Société Philomatique de Paris (1820) pp.156-165.
[Ho]	A.S. Householder, "Unitary triangularization of a nonsymmetric matrix" J. Assoc. Comp. Mach., 5 (1958) pp. 339–342 MR0111128 Zbl 0121.33802
[Se]	Serret, Joseph A.,"Cours d'algèbre supérieure", Tome I,Paris, Gauthier-Villars (1877) [4]