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| In mathematics, '''Budan's theorem''', named after [[François Budan de Boislaurent|F. D. Budan]], is a little-known theorem for computing an upper bound on the number of real roots a polynomial has inside an open interval by counting the number of '''sign variations''' or '''sign changes''' in the [[sequence]]s of coefficients. Since 1836, the statement of Budan's theorem has been replaced in the literature by the statement of an equivalent theorem by [[Joseph Fourier|Fourier]], and the latter has been referred to under various names, including Budan's. However, it is only the statement of Budan's theorem that forms the basis of the fastest method for the isolation of the real roots of polynomials and hence it deserves its own place in the history of mathematics.
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| ==Sign variation==
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| :Let <math>c_0, c_1, c_2, \ldots </math> be a finite or infinite sequence of real numbers. Suppose <math>l < r</math> and the following conditions hold:
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| # If <math>r = l + 1</math> the numbers <math>c_l</math> and <math>c_r</math> have opposite signs.
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| # If <math>r \ge l + 2</math> the numbers <math>c_{l+1}, \ldots, c_{r-1}</math> are all zero and the numbers <math>c_l</math> and <math>c_r</math> have opposite signs.
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| : This is called a ''sign variation'' or ''sign change'' between the numbers <math>c_l</math> and <math>c_r</math>.
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| : When dealing with the polynomial <math>p(x)</math> in one variable, one defines the number of ''sign variations of <math>p(x)</math>'' as the number of sign variations in the sequence of its coefficients.
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| ==Budan's theorem==
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| The following statement of Budan's theorem had disappeared from the literature (for about 150 years) due to its equivalence to the statement of [[Budan's theorem#Fourier's theorem|Fourier's theorem]].
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| === Statement of Budan's theorem === | |
| Given an equation in <math>x</math>, <math>p(x) = 0</math> of degree <math>n > 0</math> it is possible to make two substitutions <math>x \leftarrow x + l</math> and <math>x \leftarrow x + r</math> where <math>l</math> and <math>r</math> are real numbers so that <math>l < r</math>. If <math>v_l</math> and <math>v_r</math> are the [[Budan's theorem#Sign variation|sign variations]] in the sequences of the coefficients of <math>p(x+l)</math> and <math>p(x+r)</math> respectively then, provided <math>p(r) \ne 0</math>, the following applies:
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| #The polynomial <math>p(x+l)</math> cannot have fewer [[Budan's theorem#Sign variation|sign variations]] than those of <math>p(x+r)</math>. In short <math>v_l \ge v_r</math>
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| #The number <math>\rho</math> of the real roots of the equation <math>p(x) = 0</math> located in the open interval <math>(l,r)</math> can never be more than the number of [[Budan's theorem#Sign variation|sign variations]] lost in passing from the transformed polynomial <math>p(x+l)</math> to the transformed polynomial <math>p(x+r)</math>. In short, <math>\rho \le v_l - v_r</math>
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| #When the number <math>\rho</math> of the real roots of the equation <math>p(x) = 0</math> located in the open interval <math>(l,r)</math> is strictly less than the number of the [[Budan's theorem#Sign variation|sign variations]] lost in passing from the transformed polynomial <math>p(x+l)</math> to the transformed polynomial <math>p(x+r)</math> then the difference is always an even number. In short, <math>\rho = v_l - v_r -2\lambda</math> where <math>\lambda </math> ∈ <math>\mathbb{Z}_+</math>.
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| It should also be mentioned that making use of the substitutions <math>x \leftarrow x + l</math> and <math>x \leftarrow x + r</math>, the exact number of real roots in the interval <math>(l,r)</math> can be found in only two cases:
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| #If there is no sign variation loss, then there are no real roots in the interval <math>(l,r)</math>.
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| #If there is one sign variation loss, then there is exactly one real root in the interval <math>(l,r)</math>. The inverse statement does not hold in this case.
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| === Examples of Budan's theorem ===
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| 1. Given the polynomial <math>p(x)=x^3 -7x + 7 </math> and the open interval <math>(0,2)</math>, the substitutions <math>x \leftarrow x + 0</math> and <math>x \leftarrow x + 2</math> can be made. The resulting polynomials and the respective [[Budan's theorem#Sign variation|sign variations]] are:
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| : <math>p(x+0)=(x+0)^3 -7(x+0) + 7\Rightarrow p(x+0)=x^3 -7x + 7 , v_0=2</math>
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| : <math>p(x+2)=(x+2)^3 -7(x+2) + 7\Rightarrow p(x+2)=x^3+6x+5x+1, v_2=0</math>
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| Thus, from Budan's theorem <math>\rho \le v_0 - v_2 = 2 </math>. Therefore, the polynomial <math>p(x)</math> has either two or no real roots in the open interval <math>(0,2)</math>, a case that must be further investigated.
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| 2. Given the same polynomial <math>p(x)=x^3 -7x + 7 </math> and the open interval <math>(0,1)</math> the substitutions <math>x \leftarrow x + 0</math> and <math>x \leftarrow x + 1</math> can be made. The resulting polynomials and the respective [[Budan's theorem#Sign variation|sign variations]] are:
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| : <math>p(x+0)=(x+0)^3 -7(x+0) + 7\Rightarrow p(x+0)=x^3 -7x + 7 , v_0=2</math>
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| : <math>p(x+1)=(x+1)^3 -7(x+1) + 7\Rightarrow p(x+1)=x^3+3x-4x+1, v_2=2</math>
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| By Budan's theorem <math>\rho = v_0 - v_2 = 0 </math>, i.e. there are no real roots in the open interval <math>(0,1)</math>.
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| The last example indicates the main use of Budan's theorem as a [[Budan's theorem#Early applications of Budan's theorem|"no roots test"]].
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| ==Fourier's theorem==
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| The statement of '''Fourier's theorem (for Polynomials)''' which also appears as '''Fourier–Budan theorem''' or as the '''Budan–Fourier theorem''' or just as '''Budan's theorem''' can be found in almost all texts and articles on the subject.
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| ===Fourier's sequence===
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| Given an equation in <math>x</math>, <math>p(x) = 0</math> of degree <math>n > 0</math> , the '''Fourier sequence''' of <math>p(x)</math>, <math>F_\text{seq}(x)</math>, is defined as the sequence of the <math> n + 1 </math> functions <math>p(x), p^{(1)}(x),\ldots,p^{(n)}(x)</math> where <math>p^{(i)}</math> is the ith derivative of <math>p(x)</math>.Thus, <math>F_\text{seq}(x)=\big\{ p(x), p^{(1)}(x),\ldots,p^{(n)}(x)\big\}</math>
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| ====Example of Fourier's sequence====
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| The Fourier sequence of the polynomial <math>p(x)=x^3 -7x + 7 </math> is <math>F_\text{seq}(x)=\big\{x^3 -7x + 7,3x^2-7,6x,6\big\}</math>.
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| ===Statement of Fourier's theorem===
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| Given the polynomial equation <math>x</math>, <math>p(x) = 0</math> of degree <math>n > 0 </math> with real coefficients and its corresponding Fourier sequence <math>F_\text{seq}(x)=\big\{ p(x), p^{(1)}(x),\ldots,p^{(n)}(x)\big\}</math>, <math> x </math> can be replaced
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| by any two real numbers <math>l,r</math> <math>(l<r)</math>. If the two resulting arithmetic sequences are represented by <math>F_\text{seq}(l)</math> and <math>F_\text{seq}(r)</math> respectively, and their corresponding [[Budan's theorem#Sign variation|sign variations]] by <math>v_l, v_r</math>, then, provided <math>p(r) \ne 0</math>, the following applies:
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| #The sequence <math>F_\text{seq}(l)</math> cannot present fewer [[Budan's theorem#Sign variation|sign variations]] than the sequence <math>F_\text{seq}(r)</math>. In short, <math>v_l \ge v_r</math>
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| #The number <math>\rho</math> of the real roots of the equation <math>p(x) = 0</math> located in the open interval <math>(l,r)</math> can never be more than the number of [[Budan's theorem#Sign variation|sign variations]] lost in passing from the sequence <math>F_\text{seq}(l)</math> to the sequence <math>F_\text{seq}(r)</math>. In short, <math>\rho \le v_l - v_r</math>
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| #When the number <math>\rho</math> of the real roots of the equation <math>p(x) = 0</math> located in the open interval <math>(l,r)</math> is strictly less than the number of the [[Budan's theorem#Sign variation|sign variations]] lost in passing from the sequence <math>F_\text{seq}(l)</math> to the sequence <math>F_\text{seq}(r)</math> then the difference is always an even number. In short, <math>\rho = v_l - v_r -2\lambda</math> where <math>\lambda \in \mathbb{Z}_+</math>
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| ====Example of Fourier's theorem====
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| Given the previously mentioned polynomial <math>p(x)=x^3 -7x + 7 </math> and the open interval <math>(0,2)</math>, the following finite sequences and the corresponding [[Budan's theorem#Sign variation|sign variations]] can be computed:
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| : <math>F_\text{seq}(0)=\big\{7,-7,0,6\big\},v_0=2</math>
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| : <math>F_\text{seq}(2)=\big\{1,5,12,6\big\},v_2=0</math>
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| Thus, from Fourier's theorem <math>\rho \le v_0 - v_2 = 2 </math>. Therefore, the polynomial <math>p(x)</math> has either two or no real roots in the open interval <math>(0,2)</math>, a case which should be further investigated.
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| ==Historical background==
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| In the beginning of the 19th century [[François Budan de Boislaurent|F. D. Budan]] and [[Joseph Fourier|J. B. J. Fourier]] presented two different (but equivalent) theorems which enable us to determine the maximum possible number of real roots that an equation has within a given interval.
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| Budan's statement can hardly be found in the bibliography. Instead, it was replaced by Fourier's statement which is usually referred to as Fourier's theorem, or as Fourier–Budan or as Budan–Fourier or even as Budan's theorem. The actual statement of Budan's theorem appeared in 1807 in the memoir "Nouvelle méthode pour la résolution des équations numériques",<ref name=nmethode>{{cite book|last=Budan|first=François D.|title=Nouvelle méthode pour la résolution des équations numériques|year=1807|publisher=Courcier|location=Paris|url=http://books.google.com/books/about/Nouvelle_méthode_pour_la_résolution_de.html?id=VyMOAAAAQAAJ&redir_esc=y}}</ref> whereas Fourier's theorem was first published in 1820 in the "Bulletin des Sciences, par la Société Philomatique de Paris".<ref name=Fourier>{{cite journal|last=Fourier|first=Jean Baptiste Joseph|title=Sur l'usage du théorème de Descartes dans la recherche des limites des racines|year=1820|journal=Bulletin des Sciences, par la Société Philomatique de Paris|pages=156–165|url=http://ia600309.us.archive.org/22/items/bulletindesscien20soci/bulletindesscien20soci.pdf}}</ref> Due to the importance of these two theorems, there was a great controversy regarding priority rights; on this see <ref name=BF>{{cite journal|last=Akritas|first=Alkiviadis G.|title=On the Budan–Fourier Controversy|url=http://dl.acm.org/citation.cfm?id=1089243|journal=ACM-SIGSAM Bulletin|year=1981|volume=15|number=1|pages=8–10}}</ref><ref name=Reflections>{{cite journal|last=Akritas|first=Alkiviadis G.|title=Reflections on a Pair of Theorems by Budan and Fourier|url=http://www.jstor.org/stable/2690097|year=1982|journal=Mathematics Magazine|volume=55|number=5|pages=292–298}}</ref> and especially Arago's book<ref>{{citation|last=Arago|first=François|title=Biographies of distinguished scientific men|year=1859|publisher=Ticknor and Fields (English Translation)|location=Boston|url=http://books.google.com/books/about/Biographies_of_distinguished_scientific_men_de.html?id=xGgSAAAAIAAJ&redir_esc=y}}</ref> p. 383.
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| === Early applications of Budan's theorem ===
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| In "Nouvelle méthode pour la résolution des équations numériques",<ref name="nmethode"/> Budan himself used his theorem to compute the roots of any polynomial equation by calculating the decimal digits of the roots. More precisely, Budan used his theorem as a "'''no roots test'''", which can be stated as follows: if the polynomials <math>p(x+a)</math> and <math>p(x +a+ 1)</math> have (in the sequence of their coefficients) the same number of [[Budan's theorem#Sign variation|sign variations]] then it follows (from Budan's theorem) that <math>p(x)</math> has no real roots in the interval <math>(a,a+1)</math>.
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| Furthermore, in his book,<ref name="nmethode"/> p. 37, Budan presents, independently of his theorem, a "'''0_1 roots test'''", that is a criterion for determining whether a polynomial has any roots in the interval (0,1). This test can be stated as follows:
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| Perform on <math>p(x)</math> the substitution <math>x \longleftarrow \frac{1}{x +1} </math> and count the number of [[Budan's theorem#Sign variation|sign variations]] in the sequence of coefficients of the transformed polynomial; this number gives an ''upper bound'' on the number of real roots <math>p(x)</math> has inside the open interval <math>(0,1)</math>. More precisely, the number <math>\rho_{01}(p)</math> of real roots in the open interval <math>(0,1)</math> — multiplicities counted — of the polynomial <math>p(x) \in \mathbb{R}[x]</math>, of degree <math>deg(p)</math>, is bounded above by the number of [[Budan's theorem#Sign variation|sign variations]] <math>var_{01}(p)</math>, where
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| :<math>var_{01}(p) = var((x+1)^{deg(p)}p(\frac{1}{x+1}))</math>
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| and <math>var_{01}(p) \ge \rho_{01}(p)</math>. As in the case of [[Descartes' rule of signs]] if <math>var_{01}(p)=0</math> it follows that <math>\rho_{01}(p)=0</math> and if <math>var_{01}(p)=1</math> it follows that <math>\rho_{01}(p)=1</math>.
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| This test (which is a special case of the more general [[Vincent's theorem#The Alesina-Galuzzi "a_b roots test"|Alesina-Galuzzi "'''a_b roots test'''"]]) was subsequently used by [[J. V. Uspensky|Uspensky]] in the 20th century.<ref name="akritas">{{cite book|last=Akritas|first=Alkiviadis G.|title=There is no "Uspensky's Method"|url=http://dl.acm.org/citation.cfm?id=32457|year=1986|publisher=In: Proceedings of the fifth ACM Symposium on Symbolic and Algebraic Computation (SYMSAC '86, Waterloo, Ontario, Canada), pp. 88–90}}</ref> ([[J. V. Uspensky|Uspensky]],<ref name=Uspensky>{{cite book|last=Uspensky|first=James Victor|title=Theory of Equations|year=1948|publisher=McGraw–Hill Book Company|location=New York|url=http://www.google.com/search?q=uspensky+theory+of+equations&btnG=Search+Books&tbm=bks&tbo=1}}</ref> pp. 298–303, was the one who kept [[Budan's theorem#Vincent's theorem (1834 and 1836)|Vincent's theorem]] alive carrying the torch (so to speak) from Serret.<ref name=Serret>{{cite book|last=Serret|first=Joseph A.|title=Cours d'algèbre supérieure. Tome I|year=1877|publisher=Gauthier-Villars|url=http://archive.org/details/coursdalgbresu01serruoft}}</ref>)
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| Bourdon,<ref>{{cite book|last=Bourdon|first=Louis Pierre Marie|title=Éléments d'Algèbre|year=1831|publisher=Bachelier Père et Fils (6th edition)|location=Paris|url=http://archive.org/details/elementsalgebra00bourgoog}}</ref> in the last chapter of his 1831 Algebra (6th edition), pp. 717–760, combined Budan's theorem and [[Joseph Louis Lagrange|Lagrange's]] [[Joseph Louis Lagrange#Continued fractions|continued fraction method]] for approximating real roots of polynomials and, thus, gave a preview of Vincent's method, without actually giving credit to him. As Vincent mentions in the very first sentence of his 1834<ref name=paper_1834>{{cite journal|last=Vincent|first=Alexandre Joseph Hidulph|title=Mémoire sur la résolution des équations numériques|url=http://gallica.bnf.fr/ark:/12148/bpt6k57787134/f4.image.r=Agence%20Rol.langEN|journal=Mémoires de la Société Royale des Sciences, de l' Agriculture et des Arts, de Lille|year=1834|pages=1–34}}</ref> and 1836<ref name=paper_1836>{{cite journal|last=Vincent|first=Alexandre Joseph Hidulph|title=Sur la résolution des équations numériques|url=http://www-mathdoc.ujf-grenoble.fr/JMPA/PDF/JMPA_1836_1_1_A28_0.pdf|journal=Journal de Mathématiques Pures et Appliquées|volume=1|year=1836|pages=341–372}}</ref> papers, Bourdon used (in his book) a joint presentation of theirs.
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| === Disappearance of Budan's theorem ===
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| Budan's theorem forms the basis for [[Budan's theorem#Vincent's theorem (1834 and 1836)|Vincent's theorem]] and Vincent's (exponential) method for the isolation of the real roots of polynomials. Therefore, there is no wonder that Vincent in both of his papers of 1834<ref name="paper_1834"/> and 1836<ref name="paper_1836"/> states Budan's theorem and contrasts it with the one by Fourier. Vincent was the last author in the 19th century to state Budan's theorem in its original form.
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| Despite the fact that Budan's theorem was of such great importance, the appearance of [[Sturm's theorem]] in 1827 gave it (and Vincent's theorem) the death blow. Sturm's theorem solved the [[Vincent's theorem#Real root isolation methods derived from Vincent's theorem|real root isolation problem]], by defining the precise number of real roots a polynomial has in a real open interval (a, b); moreover, Sturm himself,<ref name=Sturm>{{cite journal|last=Hourya|first=Benis-Sinaceur|title=Deux moments dans l'histoire du Théorème d'algèbre de Ch. F. Sturm|journal= Revue d'histoire des sciences|year=1988|volume=41|number=2|pages=99–132|url=http://www.persee.fr/web/revues/home/prescript/article/rhs_0151-4105_1988_num_41_2_4093}}</ref> p. 108, acknowledges the great influence [[Budan's theorem#Fourier's theorem|Fourier's theorem]] had on him: « C'est en m'appuyant sur les principes qu'il a posés, et en imitant ses démonstrations, que j'ai trouvé les nouveaux théorèmes que je vais énoncer. » which translates to «It is by relying upon the principles he has laid out and by imitating his proofs that I have found the new theorems which I am about to announce.» . Because of the above, the theorems by Fourier and Sturm appear in almost all the books on the theory of equations and Sturm's method for computing the real roots of polynomials has been the only one widely known and used ever since – up to about 1980, when it was replaced (in almost all [[computer algebra system]]s) by [[Vincent's theorem#Real root isolation methods derived from Vincent's theorem|methods derived from Vincent's theorem]], the fastest one being the [[Vincent's theorem#Vincent–Akritas–Strzeboński (VAS, 2005)|Vincent–Akritas–Strzeboński]] (VAS) method.<ref name=VAS>{{cite journal|last=Akritas|first=Alkiviadis G.|coauthors=A.W. Strzeboński, P.S. Vigklas|title=Improving the performance of the continued fractions method using new bounds of positive roots|journal=Nonlinear Analysis: Modelling and Control|year=2008|volume=13|pages=265–279|url=http://www.lana.lt/journal/30/Akritas.pdf}}</ref>
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| Consequently Budan's theorem (but not his name) was pushed into oblivion. Referentially, in [[Joseph Alfred Serret|Serret]]'s book<ref name="Serret"/> there is section 121 (p. 266) on Budan's theorem but the statement is the one due to Fourier, because, as the author explains in the footnote of p. 267 , the two theorems are equivalent and Budan had clear priority. To his credit, Serret included in his Algebra,<ref name="Serret"/> pp 363–368, [[Budan's theorem#Vincent's theorem (1834 and 1836)|Vincent's theorem]] along with its proof and directed all interested readers to Vincent's papers for examples on how it is used. Serret was the last author to mention [[Budan's theorem#Vincent's theorem (1834 and 1836)|Vincent's theorem]] in the 19th century.
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| === Comeback of Budan's theorem ===
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| Budan's theorem reappeared, after almost 150 years, in Akritas' Ph.D. Thesis "Vincent's Theorem in Algebraic Manipulation", North Carolina State University, USA, 1978, and in several publications that resulted from that dissertation.<ref name="BF"/><ref name="Reflections"/> Akritas found the statement of Budan's theorem in Vincent's paper of 1836,<ref name="paper_1836"/> which was made available to him through the efforts of a librarian in the Library of the [[University of Wisconsin–Madison|University of Wisconsin–Madison, USA]].
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| ==Equivalence between the theorems by Budan and Fourier==
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| Budan's theorem is equivalent to the one by Fourier. This equivalence is obvious from the fact that, given the polynomial <math>p(x)</math> of degree <math>n > 0 </math>, the <math>n+1</math> terms of the Fourier sequence <math>F_\text{seq}(a)</math> (obtained by substituting <math>x \leftarrow a</math> in <math>F_\text{seq}(x)</math>) have the same signs with (and are proportional to) the corresponding coefficients of the polynomial <math>p(x+a)=\sum_{i=1}^n \frac{p^{(i)}(a)}{i!}\ x^i</math>, obtained from [[Taylor series|Taylor's expansion theorem]].
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| As Alesina and Galuzzi point out in Footnote 9, p. 222 of their paper,<ref>{{cite journal|last=Alesina|first=Alberto|coauthor=Massimo Galuzzi|title=A new proof of Vincent's theorem|url=http://retro.seals.ch/cntmng?type=pdf&rid=ensmat-001:1998:44::149&subp=hires|journal=L'Enseignement Mathématique|year=1998|volume=44|number=3-4|pages=219–256}}</ref> the controversy over priority rights of Budan or Fourier is rather pointless from a modern point of view. The two authors think that Budan has an "amazingly modern understanding of the relevance of reducing the algorithm (his own word) to translate a polynomial by <math>x \leftarrow x+p</math>, where <math>p</math> is an integer, to simple additions".
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| Despite their equivalence, the two theorems are quite distinct concerning the impact they had on the [[Vincent's theorem#Real root isolation methods derived from Vincent's theorem|isolation of the real roots]] of polynomials with rational coefficients. To wit:
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| * Fourier's theorem led Sturm to his theorems and [[Sturm's theorem|method]],<ref name="Sturm"/> whereas
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| * Budan's theorem is the basis of the [[Vincent's theorem#Vincent–Akritas–Strzeboński (VAS, 2005)|Vincent–Akritas–Strzeboński]] (VAS) method.<ref name="VAS"/>
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| ''[[Xcas]] is a [[computer algebra system]] where Sturm's method and VAS are both implemented and can be compared; to do so use the functions '''realroot(poly)''' and '''time(realroot(poly))'''. By default, to isolate the real roots of poly '''realroot''' uses the VAS method; to use Sturm's method write '''realroot(sturm, poly)'''. See also the [[Budan's theorem#External links|External links]] for two applications that do the same thing: one for Android devices by A. Berkakis and another one for the Apple devices iPhone/iPod/iPad by S. Kehagias.''
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| ==The most significant application of Budan's theorem==
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| Vincent's (exponential) method for the [[Vincent's theorem#Real root isolation methods derived from Vincent's theorem|isolation of the real roots]] of polynomials (which is based on Vincent's theorem of 1834 and 1836)<ref name="paper_1834"/><ref name="paper_1836"/> is the most significant application of Budan's theorem. Moreover, it is the most representative example of the importance of the statement of Budan's theorem. As explained below, knowing the statement of Fourier's theorem did not help [[J. V. Uspensky|Uspensky]] realize that there are no roots of <math>p(x)</math> in the open interval <math>(a, a+1)</math> if <math>p(x + a)</math> and <math>p(x + a + 1)</math> have the same number of [[Budan's theorem#Sign variation|sign variations]] in the sequence of their coefficients (see,<ref name="Uspensky"/> pp. 127–137).
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| ===Vincent's theorem (1834 and 1836)===
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| If in a polynomial equation with rational coefficients and without multiple roots, one makes successive transformations of the form
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| : <math>x = a + \frac{1}{x'},\quad x' = b + \frac{1}{x''},\quad x'' = c + \frac{1}{x'''}, \ldots</math>
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| where ''a'', ''b'', and ''c'' are any positive numbers greater than or equal to one, then after a number of such transformations, the resulting transformed equation either has zero [[Budan's theorem#Sign variation|sign variations]] or it has a single sign variation. In the first case there is no root, whereas in the second case there is a single positive real root. Furthermore, the corresponding root of the proposed equation is approximated by the finite continued fraction:<ref name="paper_1834"/><ref name="paper_1836"/><ref name=paper_1838>{{cite journal|last=Vincent|first=Alexandre Joseph Hidulph|title=Addition à une précédente note relative à la résolution des équations numériques|url=http://math-doc.ujf-grenoble.fr/JMPA/PDF/JMPA_1838_1_3_A19_0.pdf|journal=Journal de Mathématiques Pures et Appliquées|volume=3|year=1838|pages=235–243}}</ref>
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| : <math>a + \cfrac{1}{b + \cfrac{1}{c + \cfrac{1}{\ddots}}} </math>
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| Finally, if infinitely many numbers satisfying this property can be found, then the root is represented by the (infinite) corresponding continuous fraction.
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| The above statement is an exact translation of the theorem found in Vincent's original papers;<ref name="paper_1834"/><ref name="paper_1836"/><ref name="paper_1838"/> for a clearer understanding see the remarks in the Wikipedia article [[Vincent's theorem#Vincent's theorem: Continued fractions version (1834 and 1836)|Vincent's theorem]]
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| ===Vincent's implementation of his own theorem===
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| [[File:Vincent method.jpg|thumb|Vincent's search for a root (applying Budan's theorem)|right|450px]]
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| Vincent uses Budan's theorem exclusively as a [[Budan's theorem#Early applications of Budan's theorem|"no roots test"]] to locate where the roots lie on the ''x''-axis (to compute the quantities <math>a,b,c ,\ldots</math> of his [[Budan's theorem#Vincent's theorem (1834 and 1836)|theorem]]); that is, to find the integer part of a root Vincent performs successively substitutions of the form <math>x \leftarrow x + 1</math> and stops only when the polynomials <math>p(x)</math> and <math>p(x + 1)</math> differ in the number of [[Budan's theorem#Sign variation|sign variations]] in the sequence of their coefficients (i.e when the number of [[Budan's theorem#Sign variation|sign variations]] of <math>p(x + 1)</math> is decreased).<ref name="paper_1834"/><ref name="paper_1836"/>
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| See the corresponding diagram where the root lies in the interval <math>(5,6)</math>. Since in general the location of the root is not known in advance, the root can be found (with the help of Budan's theorem) only by this decrease in the number of [[Budan's theorem#Sign variation|sign variations]]; that is, the polynomial <math>p(x+6)</math> has fewer [[Budan's theorem#Sign variation|sign variations]] than the polynomial <math>p(x+5)</math>. Vincent then easily obtains a first continued fraction approximation to this root as <math>x = 5 +\frac{1}{x'}</math> as stated in his theorem. Vincent performs those, and only those, transformations that are described in his theorem.
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| ===Uspensky's implementation of Vincent's theorem===
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| :According to Alexei Uteshev<ref name=Descartes>{{cite book|last=Akritas|first=Alkiviadis G.|title=There is no "Descartes' method"|url=http://books.google.com/books?id=SJR2ybQdZFgC&lpg=PR1&pg=PR1#v=onepage&q&f=false|year=2008|publisher=In: M.J.Wester and M. Beaudin (Eds), Computer Algebra in Education, AullonaPress, USA, pp. 19–35}}</ref> of St. Petersburg University, Russia, [[J. V. Uspensky|Uspensky]] came upon the statement (and proof) of [[Budan's theorem#Vincent's theorem (1834 and 1836)|Vincent's theorem]] in the 20th century in Serret's Algebra,<ref name="Serret"/> pp 363–368, which means that he was not aware of the statement of Budan's theorem (because Serret included in his book Fourier's theorem). Moreover, this means that [[J. V. Uspensky|Uspensky]] never saw Vincent's papers of 1834<ref name="paper_1834"/> and 1836,<ref name="paper_1836"/> where Budan's theorem is stated and Vincent's method is explained with several examples (because Serret directed all interested readers to Vincent's papers for examples on how the theorem is used). Therefore, in the preface of his book that came out in 1949,<ref name="Uspensky"/> [[J. V. Uspensky|Uspensky]] erroneously claimed that, based on [[Budan's theorem#Vincent's theorem (1834 and 1836)|Vincent's theorem]], he had discovered a method for isolating the real roots "much superior in practice to that based on Sturm's Theorem". [[J. V. Uspensky|Uspensky]]'s statement is erroneous because, since he is not using Budan's theorem, he is isolating the real roots doing twice the amount of work done by Vincent (see,<ref name="Uspensky"/> pp. 127–137). Despite this misunderstanding, kudos to [[J. V. Uspensky|Uspensky]] for keeping [[Budan's theorem#Vincent's theorem (1834 and 1836)|Vincent's theorem]] alive.
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| [[File:Uspensky attempt.jpg|thumb|Uspensky's search for a root (not applying Budan's theorem)|450px]]
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| [[J. V. Uspensky|Uspensky]] does not know Budan's theorem and, hence, he cannot use it as a [[Budan's theorem#Early applications of Budan's theorem|"no roots test"]]. So, for him it does not suffice that <math> p(x + 1) </math> has the same number of [[Budan's theorem#Sign variation|sign variations]] as <math>p(x)</math> in order to conclude that <math>p(x)</math> has no roots inside <math>(0,1)</math>; to make sure, he also performs the redundant substitution (Budan's [[Budan's theorem#Early applications of Budan's theorem|"0_1 roots test"]]) <math>x \leftarrow \frac{1}{1+x}</math> in <math>p(x)</math>, which unfailingly results in a polynomial with no [[Budan's theorem#Sign variation|sign variations]] and hence no positive roots.
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| [[J. V. Uspensky|Uspensky]] uses the information obtained from both the needed transformation <math>x \leftarrow x + 1</math> and the not needed one <math>x \leftarrow \frac{1}{1+x}</math> to realize that <math>p(x)</math> has no roots in the interval <math>(0,1)</math>. In other words, searching for a root, Uspensky advances as illustrated in the corresponding figure.
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| [[J. V. Uspensky|Uspensky]]'s transformations are not the ones described in [[Budan's theorem#Vincent's theorem (1834 and 1836)|Vincent's theorem]], and consequently, his transformations take twice as much computation time as the ones needed for Vincent's method.<ref name="akritas"/><ref name="Descartes"/>
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| ==See also==
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| *[[Properties of polynomial roots]]
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| *[[Root-finding algorithm]]
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| *[[Vieta's formulas]]
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| *[[Newton's method]]
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| ==References==
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| {{reflist}}
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| ==External links==
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| * Berkakis, Antonis: RealRoots, a free App for Android devices to compare Sturm's method and VAS
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| * https://play.google.com/store/apps/details?id=org.kde.necessitas.berkakis.realroots
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| * Budan, F.: extended Biography http://www-history.mcs.st-andrews.ac.uk/Biographies/Budan_de_Boislaurent.html
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| * Encyclopedia of Mathematics http://www.encyclopediaofmath.org/index.php/Budan-Fourier_theorem
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| * Kehagias, Spyros: RealRoots, a free App for iPhone, iPod Touch and iPad to compare Sturm's method and VAS http://itunes.apple.com/gr/app/realroots/id483609988?mt=8
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| [[Category:Mathematical theorems]]
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