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| In [[algebra]], a '''Bring radical''' or '''ultraradical''' of a [[complex number]] ''a'' is a root of the polynomial
| | Andrew Berryhill is what his wife enjoys to call him and he totally digs that name. For a while I've been in Alaska but I will have to move in a yr or two. Credit authorising is how he makes cash. It's not a typical factor but what I like performing is to climb but I don't have the time lately.<br><br>Here is my web blog: free online tarot card readings ([http://www.prograd.uff.br/novo/facts-about-growing-greater-organic-garden uff.br]) |
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| :<math>x^5+x+a. \,</math>
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| (The root is chosen so the radical of a [[real number]] is real, and the radical is a differentiable function of ''a'' in the complex plane, with a [[branch cut]] along the negative real line below −1. See the [[Bring radical#Bring radicals|"Bring radicals"]] section below.)
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| [[George Jerrard]] showed that some [[quintic equation]]s can be solved using [[Nth root|radicals]] and Bring radicals, which had been introduced by [[Erland Samuel Bring|Erland Bring]]. They can be used to obtain closed-form solutions of quintic equations.
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| ==Normal forms==
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| The quintic equation is rather difficult to obtain solutions for directly, with five independent coefficients in its most general form:
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| :<math>x^5 + a_4x^4 + a_3x^3 + a_2x^2 + a_1x + a_0 = 0\,</math>
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| The various methods for solving the quintic that have been developed generally attempt to simplify the quintic using [[Tschirnhaus transformation]]s to reduce the number of independent coefficients.
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| ===Principal quintic form===
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| The general quintic may be reduced into what is known as the '''principal quintic form''', with the quartic and cubic terms removed:
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| :<math>x^5 + c_2x^2 + c_1x + c_0 = 0 \,</math>
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| If the roots of a general quintic and a principal quintic are related by a quadratic Tschirnhaus transformation:
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| :<math>y_k = x_k^2 + \alpha x_k + \beta \,</math>
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| the coefficients ''α'' and ''β'' may be determined by using the [[resultant]], or by means of the [[power-sum formula]]e of the quintic roots. This leads to a system of equations in ''α'' and ''β'' consisting of a quadratic and a linear equation, and either of the two sets of solutions may be used to obtain the corresponding three coefficients of the principal quintic form.<ref name="adamchik">
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| {{cite journal
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| | last = Adamchik
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| | first = Victor
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| | year = 2003
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| | title = Polynomial Transformations of Tschirnhaus, Bring, and Jerrard
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| | journal = ACM SIGSAM Bulletin
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| | volume = 37
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| | issue = 3
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| | url = http://www.sigsam.org/bulletin/articles/145/Adamchik.pdf
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| | pages = 91}}</ref>
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| This form is used by [[Felix Klein]]'s solution to the quintic.<ref name="klein">{{cite book
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| | last = Klein
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| | first = Felix
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| | authorlink=Felix Klein
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| | title = Lectures on the Icosahedron and the Solution of Equations of the Fifth Degree
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| | year = 1888
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| | publisher = Trübner & Co.
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| | isbn = 0-486-49528-0
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| | url = http://historical.library.cornell.edu/cgi-bin/cul.math/docviewer?did=03070001&seq=7}}</ref>
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| ===Bring–Jerrard normal form <!-- This section is linked from four redirect pages: [[Bring-Jerrard form]], [[Bring-Jerrard normal form]], [[Bring–Jerrard form]], and [[Bring–Jerrard normal form]] (two of these have hyphens and two have en-dashes). -->===
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| It is possible to simplify the quintic still further and eliminate the quadratic term, producing the '''Bring–Jerrard normal form''':
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| :<math>x^5 + d_1x + d_0 = 0\,</math>
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| Using the power-sum formulae again with a cubic transformation as [[Ehrenfried Walther von Tschirnhaus|Tschirnhaus]] tried does not work, since the resulting system of equations results in a sixth-degree equation.
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| In 1796 [[Erland Samuel Bring|Bring]] found a way around this by using a quartic Tschirnhaus transformation to relate the roots of a principal quintic to those of a Bring–Jerrard quintic:
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| :<math>z_k = x^4_k + \alpha x^3_k + \beta x^2_k + \gamma x_k + \delta\,</math>
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| The extra parameter this fourth-order transformation provides allowed Bring to decrease the degrees of the other parameters. This leads to a system of five equations in six unknowns, which then requires the solution of a cubic and a quadratic equation. This method was also discovered by [[George Jerrard|Jerrard]] in 1852,<ref>{{cite book
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| | last = Jerrard
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| | first = George Birch
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| | title = An essay on the resolution of equations
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| | year = 1859
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| | publisher = Taylor and Francis
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| | location = London
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| | url = http://www.archive.org/details/essayonresolutio00jerrrich}}</ref> but it is likely that he was unaware of Bring's previous work in this area.<ref>Adamchik, pp. 92–93</ref> The full transformation may readily be accomplished using a [[computer algebra]] package such as [[Mathematica]]<ref name="qmathematica">{{cite web
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| | title = Solving the Quintic with Mathematica
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| | url = http://library.wolfram.com/examples/quintic/
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| | publisher = [[Wolfram Research]]
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| }}</ref> or [[Maple (software)|Maple]].<ref name="drociuk">
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| {{cite arXiv
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| | last = Drociuk
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| | first = Richard J.
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| | year = 2000
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| | title = On the Complete Solution to the Most General Fifth Degree Polynomial
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| | eprint = math.GM/0005026
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| | class = math.GM
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| }}</ref> As might be expected from the complexity of these transformations, the resulting expressions can be enormous, particularly when compared to the solutions in radicals for lower degree equations, taking many megabytes of storage for a general quintic with symbolic coefficients.<ref name="qmathematica"/>
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| Regarded as an algebraic function, the solutions to
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| :<math>x^5+d_1x+d_0 = 0\,</math>
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| involve two variables, ''d''<sub>1</sub> and ''d''<sub>0</sub>, however the reduction is actually to an algebraic function of one variable, very much analogous to a solution in radicals, since we may further reduce the Bring–Jerrard form. If we for instance set
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| :<math>z = {x \over \sqrt[4]{-\frac{d_1}{5}}}\,</math>
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| then we reduce the equation to the form
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| :<math>z^5 - 5z - 4t = 0\,</math>
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| which involves ''z'' as an algebraic function of a single variable ''t'', where <math>t=-(d_0/4)(-d_1/5)^{-5/4}</math>. A similar transformation suffices to reduce the equation to
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| :<math>y^5 - y + a = 0\,</math>
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| which is the form required by the Hermite-Kronecker-Brioschi method, Glasser's method, and the Cockle-Harley method of differential resolvents described below.
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| ===Brioschi normal form===
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| There is another one-parameter normal form for the quintic equation, known as '''Brioschi normal form''':
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| :<math>x^5 - 10Cx^3 + 45C^2x - C^2 = 0\,</math>
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| which can be derived by using the following rational Tschirnhaus transformation
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| : <math>z_k = \frac{\lambda + \mu y_k}{\frac{y_k^2}{C}-3}\,</math>
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| to relate the roots of a principal quintic to a Brioschi quintic. The values of the parameters <math>\lambda\,</math> and <math>\mu\,</math> may be derived by using [[polyhedral function]]s on the [[Riemann sphere]], and is related to the partition of an object of [[icosahedral symmetry]] into five objects of [[tetrahedral symmetry]].<ref name="king">{{cite book
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| | last = King
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| | first = R. Bruce
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| | title = Beyond the Quartic Equation
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| | year = 1996
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| | publisher = Birkhäuser
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| | isbn = 3-7643-3776-1
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| | pages = 131}}</ref>
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| It is to be noted that this Tschirnhaus transformation is rather simpler than the difficult one used to transform a principal quintic into Bring–Jerrard form. This normal form is used by the Doyle–McMullen iteration method and the Kiepert method. | |
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| ==Series representation==
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| A [[Taylor series]] for Bring radicals, as well as a representation in terms of [[Generalized hypergeometric function|hypergeometric function]]s can be derived as follows. The equation <math>x^5+x+a=0\,</math> can be rewritten as <math>x^5+x=-a\,</math>; by setting <math>f(x)=x^5+x\,</math>, the desired solution is <math>x=f^{-1}(-a)\,</math>.
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| The series for <math>f^{-1}\,</math> can then be obtained by [[Lagrange inversion theorem|reversion]] of the [[Taylor series]] for <math>f(x)\,</math> (which is simply <math>x+x^5\,</math>), giving:
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| :<math>f^{-1}(a) = \sum_{k=0}^\infty \binom{5k}{k} \frac{(-1)^k a^{4k+1}}{4k+1} = a - a^5 + 5 a^9 - 35 a^{13} + ...</math>
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| where the absolute values of the coefficients are sequence [[OEIS:A002294|A002294]] in the [[On-Line Encyclopedia of Integer Sequences|OEIS]]. The series confirms that <math>f^{-1}(a)\,</math> is odd. This gives
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| :<math>BR(a) = f^{-1}(-a) = -f^{-1}(a) = -a + a^5 - 5 a^9 + 35 a^{13} + ... \,</math>
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| The series converges for <math> |a|< 4/(5 \cdot \sqrt[4]{5}) \approx 0.53499 </math> and can be analytically continued in the complex plane. The above result can be written in hypergeometric form as:<ref name=qmathematica/>
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| :<math>BR(a) = -a \,\,_4F_3\left(\frac{1}{5},\frac{2}{5},\frac{3}{5},\frac{4}{5};\frac{1}{2},\frac{3}{4},\frac{5}{4};-\left(\frac{5a}{4}\right)^4\right)</math>
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| Compare with the hypergeometric functions that arise in Glasser's derivation and the method of differential resolvents below.
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| ==Solution of the general quintic==
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| We now may express the roots of any polynomial
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| :<math>x^5 + px +q\,</math>
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| in terms of the Bring radical as
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| :<math>\sqrt[4]{-\frac{p}{5}}\operatorname{BR}\left(-\frac{\sqrt[4]{\frac{5^5}{-p^5}}q}{4}\right)</math>
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| and its four [[complex conjugate|conjugate]]s. We have a reduction to the Bring–Jerrard form in terms of solvable polynomial equations, and we used transformations involving polynomial expressions in the roots only up to the fourth degree, which means inverting the transformation may be done by finding the roots of a polynomial solvable in radicals. This procedure produces extraneous solutions, but when we have found the correct ones by numerical means we can also write down the roots of the quintic in terms of square roots, cube roots, and the Bring radical, which is therefore an algebraic solution in terms of algebraic functions of a single variable — an algebraic solution of the general quintic.
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| ==Other characterizations==
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| Many other characterizations of the Bring radical have been developed, the first of which is in terms of [[elliptic modular function]]s by [[Charles Hermite]] in 1858, and further methods later developed by other mathematicians.
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| ===The Hermite–Kronecker–Brioschi characterization===
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| In 1858, Charles Hermite<ref name="hermite">
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| {{cite journal
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| | last = Hermite
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| | first = Charles
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| | year = 1858
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| | title = Sur la résolution de l'équation du cinquème degré
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| | journal = Comptes Rendus de l'Académie des Sciences
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| | volume = XLVI
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| | issue = I
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| | pages = 508–515}}</ref> published the first known solution to the general quintic equation in terms of elliptic transcendents, and at around the same time [[Francesco Brioschi]]<ref>
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| {{cite journal
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| | last = Brioschi
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| | first = Francesco
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| | year = 1858
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| | title = Sul Metodo di Kronecker per la Risoluzione delle Equazioni di Quinto Grado
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| | journal = Atti dell'i. R. Istituto Lombardo di scienze, lettere ed arti
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| | volume = I
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| | pages = 275–282}}</ref> and [[Leopold Kronecker]]<ref>
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| {{cite journal
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| | last = Kronecker
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| | first = Leopold
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| | year = 1858
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| | title = Sur la résolution de l'equation du cinquième degré, extrait d'une lettre adressée à M. Hermite
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| | journal = Comptes Rendus de l'Académie des Sciences
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| | volume = XLVI
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| | issue = I
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| | pages = 1150–1152}}</ref> came upon equivalent solutions. Hermite arrived at this solution by generalizing the well-known solution to the [[cubic equation]] in terms of [[trigonometric function]]s and finds the solution to a quintic in Bring–Jerrard form:
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| :<math>x^5 - x + a = 0\,</math>
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| into which any quintic equation may be reduced by means of Tschirnhaus transformations as has been shown. He observed that [[elliptic function]]s had an analogous role to play in the solution of the Bring–Jerrard quintic as the trigonometric functions had for the cubic. If <math>K\,</math> and <math>K'\,</math> are the periods of an [[elliptic integral]] of the first kind:
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| :<math>K = \int_0^{\frac{\pi}{2}} \frac{d\varphi}{\sqrt{1-k^2 \sin^2\varphi}}</math>
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| :<math>K' = \int_0^{\frac{\pi}{2}} \frac{d\varphi}{\sqrt{1-k'^2 \sin^2\varphi}}</math>
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| the [[nome (mathematics)|elliptic nome]] is given by:
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| :<math>q =e^{-\frac{\pi K'}{K}}\,</math>
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| and
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| :<math>k^2 + k'^2 = 1\,</math>
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| With
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| :<math>q = e^{-\frac{\pi K'}{K}} = e^{{\mathrm{i}} \pi \tau}\,</math>
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| define the two [[elliptic modular function]]s:
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| :<math>\sqrt[4]{k} = \varphi(\tau) = \sqrt{\frac{\vartheta_{10}(0; \tau)}{\vartheta_{00}(0; \tau)}}</math>
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| :<math>\sqrt[4]{k'} = \psi(\tau) = \sqrt{\frac{\vartheta_{01}(0; \tau)}{\vartheta_{00}(0; \tau)}}</math>
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| where <math>\vartheta_{00}(0; \tau)</math> and similar are Jacobi [[theta function]]s.
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| If ''n'' is a [[prime number]], we can define two values ''u'' and ''v'' as follows:
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| :<math>v = \varphi(n\tau)\,</math>
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| and
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| :<math>u = \varphi(\tau)\,</math>
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| The parameters <math>u\,</math> and <math>v\,</math> are linked by an equation of degree ''n'' + 1 known as the [[modular equation]], whose ''n'' + 1 roots are given by:
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| :<math>\epsilon\varphi(n\tau)\,</math>
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| and
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| :<math>\varphi\left(\frac{\tau + 16m}{n}\right)\,</math>
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| where ε is 1 or −1 depending on whether 2 is a [[quadratic residue]] with respect to ''n'' or not, and ''m'' is an integer modulo ''n''. For ''n'' = 5, we have the modular equation of the sixth degree:
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| :<math>u^6 - v^6 + 5u^2v^2(u^2-v^2)+4uv(1-u^4v^4)=0\,</math>
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| with six roots as shown above.
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| The modular equation of the sixth degree may be related to the Bring–Jerrard quintic by the following function of the six roots of the modular equation:
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| :<math>\Phi(\tau) = \left[\varphi(5\tau) + \varphi\left(\frac{\tau}{5}\right)\right]\left[\varphi\left(\frac{\tau+16}{5}\right) - \varphi\left(\frac{\tau + 64}{5}\right)\right]\left[\varphi\left(\frac{\tau+32}{5}\right) - \varphi\left(\frac{\tau + 48}{5}\right)\right]\,</math>
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| The five quantities <math>\Phi(\tau)\,</math>, <math>\Phi(\tau+16)\,</math>, <math>\Phi(\tau+32)\,</math>, <math>\Phi(\tau+48)\,</math>, <math>\Phi(\tau+64)\,</math> are the roots of a quintic equation with coefficients rational in <math>\varphi(\tau)\,</math>:
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| :<math>\Phi^5 - 2000\varphi^4(\tau)\psi^{16}(\tau)\Phi - 1600\sqrt{5}\varphi^3(\tau)\psi^{16}(\tau)\left[1 + \varphi^8(\tau)\right] = 0\,</math>
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| which may be readily converted into the Bring–Jerrard form by the substitution:
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| :<math>\Phi = 2\sqrt[4]{125}\varphi(\tau)\psi^4(\tau)x\,</math>
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| leading to the Bring–Jerrard quintic:
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| :<math>x^5 - x + a = 0\,</math>
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| where
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| :<math>a = \frac{2[1 + \varphi^8(\tau)]}{\sqrt[4]{5^5}\varphi^2(\tau)\psi^4(\tau)}\,</math>
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| The Hermite–Kronecker–Brioschi method then amounts to finding a value for τ that corresponds to the value of ''a'', and then using that value of τ to obtain the roots of the corresponding modular equation. To do this, let
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| :<math>A = \frac{a\sqrt[4]{5^5}}{2}</math>
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| and calculate the required elliptic modulus <math>k</math> by solving the quartic equation:
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| :<math>k^4 + A^2k^3 + 2k^2 - A^2k + 1 = 0\,</math>
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| The roots of this equation are:
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| :<math>k = \tan \frac{\alpha}{4}, \tan \frac{\alpha+2\pi}{4}, \tan \frac{\pi - \alpha}{4}, \tan \frac{3\pi - \alpha}{4} \,</math>
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| where <math>\sin \alpha = \frac{4}{A^2}\,</math><ref>{{cite book
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| | last = Davis
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| | first = Harold T.
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| | title = Introduction to Nonlinear Differential and Integral Equations
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| | year = 1962
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| | publisher = Dover
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| | isbn = 0-486-60971-5
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| | pages = 173}}</ref> (note that some important references erroneously give it as <math>\sin \alpha = \frac{1}{4A^2}\,</math>
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| <ref name="king"/><ref name="hermite"/>). Any of these roots may be used as the elliptic modulus for the purposes of the method. The value of <math>\tau</math> may be easily obtained from the elliptic modulus <math>k\,</math> by the relations given above. The roots of the Bring–Jerrard quintic are then given by:
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| :<math>x_i = \frac{\Phi(\tau + 16i)}{2\sqrt[4]{125}\varphi(\tau)\psi^4(\tau)}</math>
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| for <math>i = 0, \ldots, 4</math>.
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| It may be seen that this process uses a generalization of the [[nth root]], which may be expressed as:
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| :<math>\sqrt[n]{x} = \exp \left( {\frac{1}{n}\ln x} \right)</math>
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| or more to the point, as
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| :<math>\sqrt[n]{x} = \exp \left(\frac{1}{n}\int^x_1\frac{dt}{t}\right).</math>
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| The Hermite–Kronecker–Brioschi method essentially replaces the exponential by an elliptic modular function, and the integral <math>\int^x_1\frac{dt}{t}</math> by an elliptic integral. Kronecker thought that this generalization was a special case of a still more general theorem, which would be applicable to equations of arbitrarily high degree. This theorem, known as [[Thomae's formula]], was fully expressed by [[Hiroshi Umemura]] in 1984, who used [[Siegel modular form]]s in place of the exponential/elliptic modular function, and the integral by a [[hyperelliptic integral]].<ref>
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| {{cite conference
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| | last = Umemura
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| | first = Hiroshi
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| | year = 1984
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| | title = Resolution of algebraic equations by theta constants
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| | booktitle = Tata Lectures on Theta II
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| | pages = 3.261–3.272
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| | editor = David Mumford
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| | publisher = Birkhäuser
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| | isbn = 3-7643-3109-7}}</ref>
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| ===Glasser's derivation===
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| This derivation due to M. L. Glasser<ref>
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| {{cite arXiv
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| | last = Glasser
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| | first = M. Lawrence
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| | year = 1994
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| | title = The quadratic formula made hard: A less radical approach to solving equations
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| | arxiv = math.CA/9411224
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| }}</ref> generalizes the series method presented earlier in this article to find a solution to any [[trinomial]] equation of the form:
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| :<math>
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| x^N - x + t=0 \,\!
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| </math>
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| In particular, the quintic equation can be reduced to this form by the use of Tschirnhaus transformations as shown above. Let <math>x = \zeta^{-\frac{1}{N-1}}\,</math>, the general form becomes:
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| :<math>
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| \zeta = e^{2\pi i} + t\phi(\zeta) \,\!
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| </math>
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| where
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| :<math>
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| \phi(\zeta) = \zeta^{\frac{N}{N-1}} \,\!
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| </math>
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| A formula due to [[Joseph Louis Lagrange|Lagrange]] states that for any [[analytic function]] <math>f \,</math>, in the neighborhood of a root of the transformed general equation in terms of <math>\zeta \,</math>, above may be expressed as an [[infinite series]]:
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| :<math>
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| f(\zeta) = f(e^{2\pi {\mathrm{i}}}) + \sum^\infty_{n=1} \frac{t^n}{n!}\frac{d^{n-1}}{da^{n-1}}[f'(a)|\phi(a)|^n]_{a = e^{2\pi {\mathrm{i}}}}
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| </math>
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| If we let <math>f(\zeta) = \zeta^{-\frac{1}{N-1}}\,</math> in this formula, we can come up with the root:
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| :<math>
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| x_k = e^{-\frac{2k\pi {\rm{i}}}{N -1}} - \frac{t}{N-1}\sum^\infty_{n=0}\frac{(te^{\frac{2k\pi {\rm{i}}}{N-1}})^n}{\Gamma(n + 2)}\cdot \frac{\Gamma\left(\frac{Nn}{N-1} + 1\right)}{\Gamma\left(\frac{n}{N-1} + 1\right)} </math>
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| :<math> k=1,2, 3, \dots , N-1 \,</math>
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| By the use of the [[Gauss multiplication theorem]] the infinite series above may be broken up into a finite series of [[hypergeometric function]]s:
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| :<math>\psi_n(q) =\left(\frac{e^{\frac{2n\pi {\rm{i}}}{N-1}} t}{N-1}\right)^q N^{\frac{qN}{N-1}}\frac{\prod_{k=0}^{N-1}\Gamma\left(\frac{q}{N-1} + \frac{1 + k}{N}\right)}{\Gamma\left(\frac{q}{N-1} + 1\right)\prod^{N-2}_{k=0}\Gamma\left(\frac{q+k+2}{N-1}\right)}
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| ==\left(\frac{te^{\frac{2n\pi {\rm{i}}}{N-1}}}{N-1}\right)^q N^{\frac{qN}{N-1}}\prod_{k==2}^{N}\frac{\Gamma\left(\frac{q}{N-1}+\frac{k-1}{N}\right)}{\Gamma\left(\frac{q+k}{N-1}\right)}</math>
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| :<math>
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| x_n = e^{-\frac{2n\pi {\rm{i}}}{N-1}} - \frac{t}{(N-1)^2}\sqrt{\frac{N}{2\pi(N-1)}}\sum^{N-2}_{q=0}\psi_n(q)_{(N+1)}F_N
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| \begin{bmatrix}
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| \frac{qN+N-1}{N(N-1)}, \ldots, \frac{q+N-1}{N-1}, 1; \\[8pt]
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| \frac{q+2}{N-1}, \ldots, \frac{q+N}{N-1}, \frac{q+N-1}{N-1}; \\[8pt]
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| \left(\frac{te^{\frac{2n\pi {\rm{i}}}{N-1}} }{N-1}\right)^{N-1}N^N
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| \end{bmatrix},\quad n=1,2, 3, \dots , N-1</math>
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| :<math>
| |
| x_N = \sum_{m=1}^{N-1} \frac{t}{(N-1)^2}\sqrt{\frac{N}{2\pi(N-1)}}\sum^{N-2}_{q=0}\psi_m(q)_{(N+1)}F_N
| |
| \begin{bmatrix}
| |
| \frac{qN+N-1}{N(N-1)}, \ldots, \frac{q+N-1}{N-1}, 1; \\[8pt]
| |
| | |
| \frac{q+2}{N-1}, \ldots, \frac{q+N}{N-1}, \frac{q+N-1}{N-1}; \\[8pt]
| |
| | |
| \left(\frac{te^{\frac{2m\pi {\rm{i}}}{N-1}} }{N-1}\right)^{N-1}N^N
| |
| \end{bmatrix}</math>
| |
| | |
| and the trinomial of the form,has roots
| |
| | |
| :<math>
| |
| {}_{ax^N+bx^2 + c=0,N\equiv 1\pmod{2}} \,\!
| |
| </math>
| |
| | |
| :<math>{}_{
| |
| x_{N}=-\frac{a}{2b}\sqrt{\left(\frac{c}{b}\right)^{N-1}}{}_{N-1}F_{N-2}
| |
| \begin{bmatrix}
| |
| \frac{N+1}{2N},\frac{N+3}{2N},\cdots,\frac{N-2}{N},\frac{N-1}{N},\frac{N+1}{N},\frac{N+2}{N},\cdots,\frac{3N-3}{2N},\frac{3N-1}{2N};\\[8pt]
| |
| | |
| \frac{N+1}{2N-4},\frac{N+3}{2N-4},\cdots,\frac{N-4}{N-2},\frac{N-3}{N-2},\frac{N-1}{N-2},\frac{N}{N-2},\cdots,\frac{3N-5}{2N-4},\frac{3}{2};\\[8pt]
| |
| -\frac{a^2c^{N-2}}{4b^N\left(N-2\right)^{N-2}}
| |
| \end{bmatrix}
| |
| +\sqrt{\frac{c}{b}}{\rm{i}}{}_{N-1}F_{N-2}
| |
| \begin{bmatrix}
| |
| \frac{1}{2N},\frac{3}{2N},\cdots,\frac{N-4}{2N},\frac{N-2}{2N},\frac{N+2}{2N},\frac{N+4}{2N},\cdots,\frac{2N-3}{2N},\frac{2N-1}{2N};\\[8pt]
| |
| | |
| \frac{3}{2N-4},\frac{5}{2N-4},\cdots,\frac{2N-3}{2N-4};\\[8pt]
| |
| -\frac{a^2c^{N-2}}{4b^N\left(N-2\right)^{N-2}}
| |
| \end{bmatrix}
| |
| }</math>
| |
| | |
| :<math>{}_{
| |
| x_{N-1}=-\frac{a}{2b}\sqrt{\left(\frac{c}{b}\right)^{N-1}}{}_{N-1}F_{N-2}
| |
| \begin{bmatrix}
| |
| \frac{N+1}{2N},\frac{N+3}{2N},\cdots,\frac{N-2}{N},\frac{N-1}{N},\frac{N+1}{N},\frac{N+2}{N},\cdots,\frac{3N-3}{2N},\frac{3N-1}{2N};\\[8pt]
| |
| | |
| \frac{N+1}{2N-4},\frac{N+3}{2N-4},\cdots,\frac{N-4}{N-2},\frac{N-3}{N-2},\frac{N-1}{N-2},\frac{N}{N-2},\cdots,\frac{3N-5}{2N-4},\frac{3}{2};\\[8pt]
| |
| -\frac{a^2c^{N-2}}{4b^N\left(N-2\right)^{N-2}}
| |
| \end{bmatrix}
| |
| -\sqrt{\frac{c}{b}}{\rm{i}}{}_{N-1}F_{N-2}
| |
| \begin{bmatrix}
| |
| \frac{1}{2N},\frac{3}{2N},\cdots,\frac{N-4}{2N},\frac{N-2}{2N},\frac{N+2}{2N},\frac{N+4}{2N},\cdots,\frac{2N-3}{2N},\frac{2N-1}{2N};\\[8pt]
| |
| | |
| \frac{3}{2N-4},\frac{5}{2N-4},\cdots,\frac{2N-3}{2N-4};\\[8pt]
| |
| -\frac{a^2c^{N-2}}{4b^N\left(N-2\right)^{N-2}}
| |
| \end{bmatrix}
| |
| }</math>
| |
| | |
| :<math>{}_{
| |
| x_n=-e^{\frac{2n\pi{\rm{i}}}{N-2}}\sqrt[N-2]{\frac{b}{a}}{}_{N-1}F_{N-2}
| |
| \begin{bmatrix}
| |
| -\frac{1}{N\left(N-2\right)},-\frac{1}{N\left(N-2\right)}+\frac{1}{N},-\frac{1}{N\left(N-2\right)}+\frac{2}{N},\cdots,-\frac{1}{N\left(N-2\right)}+\frac{1}{N},\frac{N-5}{2N},-\frac{1}{N\left(N-2\right)}+\frac{N-3}{2N},-\frac{1}{N\left(N-2\right)}+\frac{N+1}{2N},-\frac{1}{N\left(N-2\right)}+\frac{N+3}{2N},\cdots,-\frac{1}{N\left(N-2\right)}+\frac{N-1}{N},;\\[8pt]
| |
| | |
| \frac{1}{N-2},\frac{2}{N-2},\cdots,\frac{2N-5}{2N-4},;\\[8pt]
| |
| -\frac{a^2c^{N-2}}{4b^N\left(N-2\right)^{N-2}}
| |
| \end{bmatrix}
| |
| +\sqrt[N-2]{\frac{b}{a}}\sum^{N-3}_{q=1}\frac{\Gamma\left(\frac{2q-1}{N-2}+q\right)}{\Gamma\left(\frac{2q-1}{N-2}+1\right)}\cdot\left(-\frac{c}{b}\sqrt[N-2]{\frac{a^2}{b^2}}\right)^q\cdot\frac{e^{\frac{2n\left(1-2q\right)}{N-2}\pi{\rm{i}}}}{q!}{}_{N-1}F_{N-2}
| |
| \begin{bmatrix}
| |
| \frac{Nq-1}{N\left(N-2\right)},\frac{Nq-1}{N\left(N-2\right)}+\frac{1}{N},\frac{Nq-1}{N\left(N-2\right)}+\frac{2}{N},\cdots,\frac{Nq-1}{N\left(N-2\right)}+\frac{N-3}{2N},\frac{Nq-1}{N\left(N-2\right)}+\frac{N+1}{2N},\cdots,\frac{Nq-1}{N\left(N-2\right)}+\frac{N-1}{N};\\[8pt]
| |
| | |
| \frac{q+1}{N-2},\frac{q+2}{N-2},\cdots,\frac{N-4}{N-2},\frac{N-3}{N-2},\frac{N-1}{N-2},\frac{N}{N-2},\cdots,\frac{q+N-2}{N-2},\frac{2q+2N-5}{2N-4};\\[8pt]
| |
| -\frac{a^2c^{N-2}}{4b^N\left(N-2\right)^{N-2}}
| |
| \end{bmatrix},n=1,2,\cdots,N-2
| |
| }</math>
| |
| | |
| A root of the equation can thus be expressed as the sum of at most ''N'' − 1 hypergeometric functions. Applying this method to the reduced Bring–Jerrard quintic, define the following functions:
| |
| | |
| :<math>
| |
| \begin{align}
| |
| F_1(t) & = \,_4F_3\left(\frac{-1}{20}, \frac{3}{20}, \frac{7}{20}, \frac{11}{20}; \frac{1}{4}, \frac{1}{2}, \frac{3}{4}; \frac{3125t^4}{256}\right) \\[6pt]
| |
| F_2(t) & = \,_4F_3\left(\frac{1}{5}, \frac{2}{5}, \frac{3}{5}, \frac{4}{5}; \frac{1}{2}, \frac{3}{4}, \frac{5}{4}; \frac{3125t^4}{256}\right) \\[6pt]
| |
| F_3(t) & = \,_4F_3\left(\frac{9}{20}, \frac{13}{20}, \frac{17}{20}, \frac{21}{20}; \frac{3}{4}, \frac{5}{4}, \frac{3}{2}; \frac{3125t^4}{256}\right) \\[6pt]
| |
| F_4(t) & = \,_4F_3\left(\frac{7}{10}, \frac{9}{10}, \frac{11}{10}, \frac{13}{10}; \frac{5}{4}, \frac{3}{2}, \frac{7}{4}; \frac{3125t^4}{256}\right)
| |
| \end{align}
| |
| </math> | |
| | |
| which are the hypergeometric functions that appear in the series formula above. The roots of the quintic are thus:
| |
| :<math>
| |
| \begin{array}{rcrcccccc}
| |
| x_1 & = & {} -tF_2(t) \\[8pt]
| |
| x_2 & = & {} -F_1(t) & + & \frac{1}{4}tF_2(t) & + & \frac{5}{32}t^2F_3(t) & + & \frac{5}{32}t^3F_4(t)\\[8pt]
| |
| | |
| x_3 & = & F_1(t) & + & \frac{1}{4}tF_2(t) & + & \frac{5}{32}t^2F_3(t) & + & \frac{5}{32}t^3F_4(t)\\[8pt]
| |
| | |
| x_4 & = & {} -{\mathrm{i}}F_1(t) & + & \frac{1}{4}tF_2(t) & - & \frac{5}{32}{\mathrm{i}}t^2F_3(t) & - & \frac{5}{32}t^3F_4(t)\\[8pt]
| |
| | |
| x_5 & = & {\mathrm{i}}F_1(t) & + & \frac{1}{4}tF_2(t) & - & \frac{5}{32}{\mathrm{i}}t^2F_3(t) & - & \frac{5}{32}t^3F_4(t)
| |
| \end{array}
| |
| </math>
| |
| | |
| This is essentially the same result as that obtained by the method of [[differential resolvent]]s developed by [[James Cockle (lawyer)|James Cockle]] and [[Robert Harley (mathematician)|Robert Harley]] in 1860.
| |
| | |
| ===The method of differential resolvents===
| |
| | |
| James Cockle<ref>
| |
| {{cite journal
| |
| | last = Cockle
| |
| | first = James
| |
| | year = 1860
| |
| | title = Sketch of a Theory of Transcendental Roots
| |
| | journal = The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science
| |
| | volume = 20
| |
| | url = http://www.archive.org/details/londonedinburghd20lond
| |
| | pages = 145–148}}</ref> and Robert Harley<ref>
| |
| {{cite journal
| |
| | last = Harley
| |
| | first = Robert
| |
| | year = 1862
| |
| | title = On the Transcendental Solution of Algebraic Equations
| |
| | journal = Quart. J. Pure Appl. Math
| |
| | volume = 5
| |
| | pages = 337–361}}</ref> developed a method for solving the quintic by means of differential equations. They consider the roots as being functions of the coefficients, and calculate a differential resolvent based on these equations. The Bring–Jerrard quintic is expressed as a function:
| |
| :<math>f(x) = x^5 - x + a\,</math>
| |
| and a function <math>\phi(a)\,</math> is to be determined such that:
| |
| | |
| :<math>f[\phi(a)] = 0\,</math>
| |
| | |
| The function <math>\phi\,</math> must also satisfy the following four differential equations:
| |
| | |
| :<math>
| |
| \begin{align}
| |
| \frac{d f[\phi(a)]}{da} = 0 \\[6pt]
| |
| \frac{d^2 f[\phi(a)]}{da^2} = 0 \\[6pt]
| |
| \frac{d^3 f[\phi(a)]}{da^3} = 0 \\[6pt]
| |
| \frac{d^4 f[\phi(a)]}{da^4} = 0
| |
| \end{align}
| |
| </math>
| |
| | |
| Expanding these and combining them together yields the differential resolvent:
| |
| | |
| :<math>
| |
| \frac{(256 - 3125a^4)}{1155}\frac{d^4\phi}{da^4} - \frac{6250a^3}{231}\frac{d^3\phi}{da^3} - \frac{4875a^2}{77}\frac{d^2\phi}{da^2} - \frac{2125a}{77}\frac{d\phi}{da} + \phi = 0
| |
| </math>
| |
| | |
| The solution of the differential resolvent, being a fourth order ordinary differential equation, depends on four [[constants of integration]], which should be chosen so as to satisfy the original quintic. This is a Fuchsian ordinary differential equation of hypergeometric type,<ref>{{cite book
| |
| | last = Slater
| |
| | first = Lucy Joan
| |
| | title = Generalized Hypergeometric Functions
| |
| | year = 1966
| |
| | publisher = Cambridge University Press
| |
| | isbn = 978-0-521-06483-5
| |
| | pages = 42–44}}</ref> whose solution turns out to be identical to the series of hypergeometric functions that arose in Glasser's derivation above.<ref name="drociuk"/>
| |
| | |
| This method may also be generalized to equations of arbitrarily high degree, with differential resolvents which are [[partial differential equation]]s, whose solutions involve hypergeometric functions of several variables.<ref>{{cite journal
| |
| | last = Birkeland
| |
| | first = Richard
| |
| | year = 1927
| |
| | title = Über die Auflösung algebraischer Gleichungen durch hypergeometrische Funktionen
| |
| | journal = Mathematische Zeitschrift
| |
| | volume = 26
| |
| | url = http://dz-srv1.sub.uni-goettingen.de/sub/digbib/loader?ht=VIEW&did=D82677
| |
| | pages = 565–578
| |
| | doi = 10.1007/BF01475474}}</ref><ref>{{cite journal
| |
| | last = Mayr
| |
| | first = Karl
| |
| | year = 1937
| |
| | title = Über die Auflösung algebraischer Gleichungssysteme durch hypergeometrische Funktionen
| |
| | journal = Monatshefte für Mathematik und Physik
| |
| | volume = 45
| |
| | pages = 280–313
| |
| | doi = 10.1007/BF01707992}}</ref>
| |
| A general formula for differential resolvents of arbitrary univariate polynomials is given by Nahay's powersum formula.
| |
| <ref>{{cite journal
| |
| | doi = 10.1155/S0161171204210602
| |
| | last = Nahay
| |
| | first = John
| |
| | year = 2004
| |
| | title = Powersum formula for differential resolvents
| |
| | journal = International Journal of Mathematics & Mathematical Sciences.
| |
| | volume = 2004| issue = 7
| |
| | pages = 365–371
| |
| }}</ref><ref>{{cite journal
| |
| | last = Nahay
| |
| | first = John
| |
| | year = 2000
| |
| | title = Linear Differential Resolvents (Doctoral dissertation), Rutgers University, Piscataway, NJ. Richard M. Cohn, advisor.
| |
| }}</ref>
| |
| | |
| ===Doyle–McMullen iteration===
| |
| | |
| In 1989, Peter Doyle and Curt McMullen derived an iteration method<ref>{{cite journal
| |
| | last = Doyle
| |
| | first = Peter
| |
| | coauthors = Curt McMullen
| |
| | year = 1989
| |
| | title = Solving the quintic by iteration
| |
| | journal = Acta Math
| |
| | volume = 163
| |
| | url = http://www.math.harvard.edu/~ctm/papers/home/text/papers/icos/icos.pdf
| |
| | pages = 151–180
| |
| | doi = 10.1007/BF02392735}}</ref> that solves a quintic in Brioschi normal form:
| |
| :<math>x^5 - 10Cx^3 + 45C^2x - C^2 = 0.\,</math>
| |
| The iteration algorithm proceeds as follows:
| |
| | |
| 1. Set <math>Z = 1 - 1728C\,</math>
| |
| 2. Compute the rational function
| |
| :: <math>T_Z(w) = w - 12\frac{g(Z,w)}{g'(Z,w)}\,</math>
| |
| :where <math>g(Z,w)\,</math> is a polynomial function given below, and <math>g'\,</math> is the [[derivative]] of <math>g(Z,w)\,</math> with respect to <math>w\,</math>
| |
| 3. Iterate <math>T_Z[T_Z(w)]\,</math> on a random starting guess until it converges. Call the [[limit point]] <math>w_1\,</math> and let <math>w_2 = T_Z(w_1)\,</math>.
| |
| 4. Compute
| |
| ::<math>\mu_i = \frac{100Z(Z-1)h(Z,w_i)}{g(Z, w_i)}\,</math>
| |
| :where <math>h(Z,w)\,</math> is a polynomial function given below. Do this for both <math>w_1\,</math> and <math>w_2 = T_Z(w_1)\,</math>.
| |
| 5. Finally, compute
| |
| ::<math>x_i = \frac{(9 + \sqrt{15}{\mathrm{i}})\mu_i + (9 - \sqrt{15}{\mathrm{i}})\mu_{3-i}}{90}</math>
| |
| :for ''i'' = 1, 2. These are two of the roots of the Brioschi quintic.
| |
| | |
| The two polynomial functions <math>g(Z,w)\,</math> and <math>h(Z,w)\,</math> are as follows:
| |
| | |
| : <math>
| |
| \begin{align}
| |
| g(Z,w) & = 91125Z^6 \\
| |
| & {} \quad {} + (-133650w^2 + 61560w - 193536)Z^5 \\
| |
| & {} \quad {} + (-66825w^4 + 142560w^3 + 133056w^2 - 61140w + 102400)Z^4 \\
| |
| & {} \quad {} + (5940w^6 + 4752w^5 + 63360w^4 - 140800w^3)Z^3 \\
| |
| & {} \quad {} + (-1485w^8 + 3168w^7 - 10560w^6)Z^2 \\
| |
| & {} \quad {} + (-66w^{10} + 440w^9)Z \\
| |
| & {} \quad {} + w^{12} \\[8pt]
| |
| h(Z,w) = & (1215w - 648)Z^4 \\
| |
| & {} \quad {} + (-540w^3 - 216w^2 - 1152w + 640)Z^3 \\
| |
| & {} \quad {} + (378w^5 - 504w^4 + 960w^3)Z^2 \\
| |
| & {} \quad {} + (36w^7 - 168w^6)Z \\
| |
| & {} \quad {} + w^9
| |
| \end{align}
| |
| </math>
| |
| | |
| This iteration method produces two roots of the quintic. The remaining three roots can be obtained by using [[synthetic division]] to divide the two roots out, producing a cubic equation. It is to be noted that due to the way the iteration is formulated, this method seems to always find two [[complex conjugate]] roots of the quintic even when all the quintic coefficients are real and the starting guess is real. This iteration method is derived by from the symmetries of the [[icosahedron]] and is closely related to the method Felix Klein describes in his book.<ref name="klein"/>
| |
| | |
| ==See also==
| |
| | |
| *[[Theory of equations]]
| |
| | |
| ==Notes==
| |
| {{reflist|2}}
| |
| | |
| ==References==
| |
| | |
| * [[Felix Klein]], ''[http://historical.library.cornell.edu/cgi-bin/cul.math/docviewer?did=03070001&seq=7 Lectures on the Icosahedron and the Solution of Equations of the Fifth Degree]'', trans. George Gavin Morrice, Trübner & Co., 1888. ISBN 0-486-49528-0.
| |
| * R. Bruce King, ''Beyond the Quartic Equation'', Birkhäuser, 1996. ISBN 3-7643-3776-1
| |
| * Harold T. Davis, ''Introduction to Nonlinear Differential and Integral Equations'', Dover, 1962, ISBN 0-486-60971-5, Chapter 6, especially Sections 20 and 21
| |
| | |
| ==External links==
| |
| * {{SpringerEOM|urlname=T/t120180|title=Tschirnhausen transformation|author=M. Hazewinkel}}
| |
| * {{MathWorld|urlname=Bring-JerrardQuinticForm|title=Bring–Jerrard Quintic Form}}
| |
| * {{MathWorld|urlname=BringQuinticForm|title=Bring Quintic Form}}
| |
| | |
| {{DEFAULTSORT:Bring Radical}}
| |
| [[Category:Equations]]
| |
| [[Category:Polynomials]]
| |