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The number of roots of an algebraic equation
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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
  
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$$f(x)=0$$
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$$f(a),f'(a),\ldots,f^{(n)}(a),$$
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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
 
  
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$$f(a),f'(a),\dots,f^{(n)}(a),$$
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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].  
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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).
 
  
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====Comments====
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As has already been noted by Serret {{Cite|Se|p. 267}}, 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}}.
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An application of the Budan–Fourier theorem in numerical analysis may be found in
 
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{{Cite|BoSc}}, where it is used in the interpolation by spline functions.
 
  
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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.
  
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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====
 
====References====
 
{|
 
{|
 
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|-
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|valign="top"|{{Ref|AlMaCh}}||valign="top"| Alexandroff, P.S., Markuschewitsch, A.I., Chintschin, A.J., "Encyclopaedia of elementary mathematics", '''2. Algebra''', Moscow-Leningrad  (1951)  pp. 331  (In Russian) {{MR|0080060}} {{ZBL|0365.00003}}
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|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]
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|-
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|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]
 +
|-
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|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]
 +
|-
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|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}}
 +
|-
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|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|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]
 
|-
 
|-
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|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}}
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|valign="top"|{{Ref|Fo}}||valign="top"| J.P.J. Fourier, "Sur l'usage du théorème 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|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}}
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|-
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|valign="top"|{{Ref|Se}}||valign="top"|Serret, Joseph A.,"Cours d'algèbre supérieure", Tome I,Paris, Gauthier-Villars (1877)
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[http://archive.org/details/coursdalgbresu01serruoft]
 
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Latest revision as of 13:01, 24 April 2012

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

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

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.

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 théorème 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]

How to Cite This Entry:
Budan-Fourier theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Budan-Fourier_theorem&oldid=24602
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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