View source for Budan-Fourier theorem

{{MSC|12Y|65T}}
{{TEX|done}}

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) {{Cite|Bu}} and J. Fourier (1820) {{Cite|Fo}}. See also the Wikipedia article [http://en.wikipedia.org/wiki/Budan's_Theorem Budan's theorem]. 

====Comments====
As per Serret{{Cite|Se}}, footnote p. 267, we alert the reader that the above statement is due to Fourier{{Cite|Fo}}. The statement of Budan's theorem can be found in the Wikipedia [http://en.wikipedia.org/wiki/Budan's_Theorem article] mentioned above. On this topic, see also {{Cite|Ak1}} and {{Cite|Ak2}}.

An application of the Budan–Fourier theorem in numerical analysis may be found in
{{Cite|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 {{Cite|Ak3}}, where it is used as a '''no roots test'''.

====References====
{|
|-
|valign="top"|{{Ref|Ak1}}||valign="top"| Akritas, Alkiviadis G.,"On the Budan–Fourier Controversy", ACM-SIGSAM Bulletin,'''15''', No. 1 (1981) pp. 8–10
[http://dl.acm.org/citation.cfm?id=1089243]
|-
|valign="top"|{{Ref|Ak2}}||valign="top"| Akritas, Alkiviadis G., "Reflections on a Pair of Theorems by Budan and Fourier", Mathematics Magazine, '''55''', No.5 (1982) pp. 292–298
[http://www.jstor.org/stable/2690097]
|-
|valign="top"|{{Ref|Ak3}}||valign="top"| Akritas, Alkiviadis G., "Vincent's Theorem of 1836: Overview and Future Research", Journal of Mathematical Sciences, '''168''', (2010) pp. 309–325
[http://www.mathnet.ru/links/03a2bad10367b8f2b6cf6e8579e189aa/znsl3571.pdf]
|-
|valign="top"|{{Ref|AlMaCh}}||valign="top"| P.S. Alexandroff, A.T. Markuschewitsch, A.J. Chintschin, "Encyclopaedia of elementary mathematics", '''2. Algebra''', Moscow-Leningrad  (1951)  pp. 331  (In Russian) {{MR|0080060}} {{ZBL|0365.00003}}
|-
|valign="top"|{{Ref|BoSc}}||valign="top"|  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) {{MR|0493059}}  {{ZBL|0319.41010}}
|-
|valign="top"|{{Ref|Bu}}||valign="top"| Budan, F. D., " Nouvelle méthode pour la résolution des équations numériques", Paris: Courcier (1807) [http://books.google.com/books/about/Nouvelle_m%C3%A9thode_pour_la_r%C3%A9solution_de.html?id=VyMOAAAAQAAJ&redir_esc=y google books]
|-
|valign="top"|{{Ref|Fo}}||valign="top"| J.P.J. Fourier, "Sur l'usage du théoréme de Descartes dans la recherche des limites des racines", [http://ia600309.us.archive.org/22/items/bulletindesscien20soci/bulletindesscien20soci.pdf Bulletin des Sciences, par la Société Philomatique de Paris]  (1820) pp.156-165.
|-
|valign="top"|{{Ref|Ho}}||valign="top"|  A.S. Householder,   "Unitary triangularization of a nonsymmetric matrix"  ''J. Assoc. Comp. Mach.'', '''5'''  (1958)  pp. 339–342   {{MR|0111128}} {{ZBL|0121.33802}}
|-
|valign="top"|{{Ref|Se}}||valign="top"|Serret, Joseph A.,"Cours d'algèbre supérieure", Tome I,Paris, Gauthier-Villars (1877)
[http://archive.org/details/coursdalgbresu01serruoft]
|-
|}