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algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...
, the Bring radical or ultraradical of a
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
 ''a'' is the unique real
root In vascular plants, the roots are the organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often below the sur ...
of the
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
: x^5 + x + a. The Bring radical of a complex number ''a'' is either any of the five roots of the above polynomial (it is thus
multi-valued In mathematics, a multivalued function, also called multifunction, many-valued function, set-valued function, is similar to a function, but may associate several values to each input. More precisely, a multivalued function from a domain to a ...
), or a specific root, which is usually chosen such that the Bring radical is real-valued for real ''a'' and is an
analytic function In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex an ...
in a neighborhood of the real line. Because of the existence of four
branch point In the mathematical field of complex analysis, a branch point of a multi-valued function (usually referred to as a "multifunction" in the context of complex analysis) is a point such that if the function is n-valued (has n values) at that point, a ...
s, the Bring radical cannot be defined as a function that is continuous over the whole
complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
, and its domain of continuity must exclude four
branch cut In the mathematical field of complex analysis, a branch point of a multi-valued function (usually referred to as a "multifunction" in the context of complex analysis) is a point such that if the function is n-valued (has n values) at that point, a ...
s. George Jerrard showed that some
quintic equation In algebra, a quintic function is a function of the form :g(x)=ax^5+bx^4+cx^3+dx^2+ex+f,\, where , , , , and are members of a field, typically the rational numbers, the real numbers or the complex numbers, and is nonzero. In other words, a q ...
s can be solved in closed form using radicals and Bring radicals, which had been introduced by Erland Bring. In this article, the Bring radical of ''a'' is denoted \operatorname(a). For real argument, it is odd, monotonically decreasing, and unbounded, with asymptotic behavior \mathrm(a) \sim -a^ for large a.


Normal forms

The quintic equation is rather difficult to obtain solutions for directly, with five independent coefficients in its most general form: :x^5 + a_4x^4 + a_3x^3 + a_2x^2 + a_1x + a_0 = 0.\, The various methods for solving the quintic that have been developed generally attempt to simplify the quintic using
Tschirnhaus transformation In mathematics, a Tschirnhaus transformation, also known as Tschirnhausen transformation, is a type of mapping on polynomials developed by Ehrenfried Walther von Tschirnhaus in 1683. Simply, it is a method for transforming a polynomial equatio ...
s to reduce the number of independent coefficients.


Principal quintic form

The general quintic may be reduced into what is known as the principal quintic form, with the quartic and cubic terms removed: :y^5 + c_2y^2 + c_1y + c_0 = 0 \, If the roots of a general quintic and a principal quintic are related by a quadratic
Tschirnhaus transformation In mathematics, a Tschirnhaus transformation, also known as Tschirnhausen transformation, is a type of mapping on polynomials developed by Ehrenfried Walther von Tschirnhaus in 1683. Simply, it is a method for transforming a polynomial equatio ...
:y_k = x_k^2 + \alpha x_k + \beta \, , the coefficients ''α'' and ''β'' may be determined by using the
resultant In mathematics, the resultant of two polynomials is a polynomial expression of their coefficients, which is equal to zero if and only if the polynomials have a common root (possibly in a field extension), or, equivalently, a common factor (over t ...
, or by means of the power sums of the roots and
Newton's identities In mathematics, Newton's identities, also known as the Girard–Newton formulae, give relations between two types of symmetric polynomials, namely between power sums and elementary symmetric polynomials. Evaluated at the roots of a monic polynomia ...
. 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. This form is used by
Felix Klein Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and group ...
's solution to the quintic.


Bring–Jerrard normal form

It is possible to simplify the quintic still further and eliminate the quadratic term, producing the Bring–Jerrard normal form: :v^5 + d_1v + d_0 = 0.\, Using the power-sum formulae again with a cubic transformation as
Tschirnhaus Ehrenfried Walther von Tschirnhaus (or Tschirnhauß, ; 10 April 1651 – 11 October 1708) was a German mathematician, physicist, physician, and philosopher. He introduced the Tschirnhaus transformation and is considered by some to have been the ...
tried does not work, since the resulting system of equations results in a sixth-degree equation. But in 1796 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: :v_k = y^4_k + \alpha y^3_k + \beta y^2_k + \gamma y_k + \delta\, . 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 Jerrard in 1852, but it is likely that he was unaware of Bring's previous work in this area. The full transformation may readily be accomplished using a
computer algebra In mathematics and computer science, computer algebra, also called symbolic computation or algebraic computation, is a scientific area that refers to the study and development of algorithms and software for manipulating mathematical expressions ...
package such as
Mathematica Wolfram Mathematica is a software system with built-in libraries for several areas of technical computing that allow machine learning, statistics, symbolic computation, data manipulation, network analysis, time series analysis, NLP, optimizat ...
or
Maple ''Acer'' () is a genus of trees and shrubs commonly known as maples. The genus is placed in the family Sapindaceae.Stevens, P. F. (2001 onwards). Angiosperm Phylogeny Website. Version 9, June 2008 nd more or less continuously updated since http ...
. 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. Regarded as an algebraic function, the solutions to :v^5+d_1v+d_0 = 0\, involve two variables, ''d''1 and ''d''0; 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 :z = \, then we reduce the equation to the form :z^5 - z + a = 0\, , which involves ''z'' as an algebraic function of a single variable a, where a=d_0(-d_1)^. This form is required by the Hermite–Kronecker–Brioschi method, Glasser's method, and the Cockle–Harley method of differential resolvents described below. An alternative form is obtained by setting u = \, so that u^5 + u + b = 0\, , where b=d_0(d_1)^. This form is used to define the Bring radical below.


Brioschi normal form

There is another one-parameter normal form for the quintic equation, known as Brioschi normal form :w^5 - 10Cw^3 + 45C^2w - C^2 = 0, which can be derived by using the rational Tschirnhaus transformation : w_k = \frac to relate the roots of a general quintic to a Brioschi quintic. The values of the parameters \lambda\, and \mu\, may be derived by using
polyhedral function Polyhedral may refer to: * Dihedral (disambiguation), various meanings *Polyhedral compound * Polyhedral combinatorics * Polyhedral cone * Polyhedral cylinder * Polyhedral convex function * Polyhedral dice * Polyhedral dual * Polyhedral formula *P ...
s on the
Riemann sphere In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers pl ...
, and are related to the partition of an object of
icosahedral symmetry In mathematics, and especially in geometry, an object has icosahedral symmetry if it has the same symmetries as a regular icosahedron. Examples of other polyhedra with icosahedral symmetry include the regular dodecahedron (the dual of the ...
into five objects of
tetrahedral symmetry 150px, A regular tetrahedron, an example of a solid with full tetrahedral symmetry A regular tetrahedron has 12 rotational (or orientation-preserving) symmetries, and a symmetry order of 24 including transformations that combine a reflection a ...
. 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.


Series representation

A
Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor serie ...
for Bring radicals, as well as a representation in terms of hypergeometric functions can be derived as follows. The equation x^5+x+a=0 can be rewritten as x^5+x=-a. By setting f(x)=x^5+x, the desired solution is x=f^(-a)=-f^(a) since f(x) is odd. The series for f^ can then be obtained by reversion of the
Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor serie ...
for f(x) (which is simply x+x^5), giving :\operatorname(a) = -f^(a) = \sum_^\infty \binom \frac = -a + a^5 - 5 a^9 + 35 a^ - 285 a^ + \cdots, where the absolute values of the coefficients form sequence A002294 in the
OEIS The On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of integer sequences. It was created and maintained by Neil Sloane while researching at AT&T Labs. He transferred the intellectual property and hosting of the OEIS to the ...
. The
radius of convergence In mathematics, the radius of convergence of a power series is the radius of the largest disk at the center of the series in which the series converges. It is either a non-negative real number or \infty. When it is positive, the power series co ...
of the series is 4/(5 \cdot \sqrt \approx 0.53499. In hypergeometric form, the Bring radical can be written as :\operatorname(a) = -a \,\,_4F_3\left(\frac,\frac,\frac,\frac;\frac,\frac,\frac;-5\left(\frac\right)^4\right). It may be interesting to compare with the hypergeometric functions that arise below in Glasser's derivation and the method of differential resolvents.


Solution of the general quintic

The roots of the polynomial :x^5 + px +q\, can be expressed in terms of the Bring radical as :\sqrt ,\operatorname\left(p^q\right) and its four conjugates. The problem is now reduced to the Bring–Jerrard form in terms of solvable polynomial equations, and using 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 gives extraneous solutions, but when the correct ones have been found by numerical means, the roots of the quintic can be written in terms of square roots, cube roots, and the Bring radical, which is therefore an algebraic solution in terms of algebraic functions (defined broadly to include Bring radicals) of a single variable — an algebraic solution of the general quintic.


Other characterizations

Many other characterizations of the Bring radical have been developed, the first of which is in terms of "elliptic transcendents" (related to
elliptic In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in ...
and
modular Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a sy ...
functions) by
Charles Hermite Charles Hermite () FRS FRSE MIAS (24 December 1822 – 14 January 1901) was a French mathematician who did research concerning number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra. Hermi ...
in 1858, and further methods later developed by other mathematicians.


The Hermite–Kronecker–Brioschi characterization

In 1858, Charles Hermite published the first known solution to the general quintic equation in terms of "elliptic transcendents", and at around the same time
Francesco Brioschi Francesco Brioschi (22 December 1824 – 13 December 1897) was an Italian mathematician. Biography Brioschi was born in Milan in 1824. He graduated from the Collegio Borromeo in 1847. From 1850 he taught analytical mechanics in the University o ...
and
Leopold Kronecker Leopold Kronecker (; 7 December 1823 – 29 December 1891) was a German mathematician who worked on number theory, algebra and logic. He criticized Georg Cantor's work on set theory, and was quoted by as having said, "'" ("God made the integers, ...
came upon equivalent solutions. Hermite arrived at this solution by generalizing the well-known solution to the
cubic equation In algebra, a cubic equation in one variable is an equation of the form :ax^3+bx^2+cx+d=0 in which is nonzero. The solutions of this equation are called roots of the cubic function defined by the left-hand side of the equation. If all of the ...
in terms of
trigonometric function In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all ...
s and finds the solution to a quintic in Bring–Jerrard form: :x^5 - x + a = 0 into which any quintic equation may be reduced by means of Tschirnhaus transformations as has been shown. He observed that
elliptic function In the mathematical field of complex analysis, elliptic functions are a special kind of meromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because they come from elliptic integrals. Originally those in ...
s had an analogous role to play in the solution of the Bring–Jerrard quintic as the trigonometric functions had for the cubic. For K and K', write: :K(k) = \int_0^ \frac\quad (the
complete elliptic integral of the first kind In integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied by Giulio Fagnano and Leonhard Euler (). Their name originates from their originally arising in ...
) :K'(k) = \int_0^ \frac where :k^2 + k'^2 = 1. Define the two "elliptic transcendents":\varphi^8(\tau)+\psi^8(\tau)=1 and \psi(\tau)=\varphi(-1/\tau). These functions are related to the
Jacobi theta functions In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. As Grassmann algebras, they appear in quantum field ...
by \varphi^2(\tau)=\vartheta_(0;\tau)/\vartheta_(0;\tau) and \psi^2(\tau)=\vartheta_(0;\tau)/\vartheta_(0;\tau).
:\varphi(\tau) = \prod_^\infty \tanh \frac=\sqrte^\prod_^\infty \frac,\quad \operatorname\tau>0 :\psi(\tau) = \prod_^\infty \tanh \frac,\quad\operatorname\tau>0 They can be equivalently defined by infinite series: :\varphi(\tau)=\sqrte^\frac,\quad \operatorname\tau >0 :\psi(\tau)=\frac,\quad\operatorname\tau >0 If ''n'' is a
prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
, we can define two values u and v as follows: :u = \varphi(n\tau) and :v = \varphi(\tau) When ''n'' is an odd prime, the parameters u and v are linked by an equation of degree ''n'' + 1 in u,When ''n'' = 2, the parameters are linked by an equation of degree 8 in u. \Omega_n(u,v)=0, known as the
modular equation In mathematics, a modular equation is an algebraic equation satisfied by ''moduli'', in the sense of moduli problems. That is, given a number of functions on a moduli space, a modular equation is an equation holding between them, or in other words ...
, whose ''n'' + 1 roots in u are given by:Some references define u=\varphi(\tau) and v=\varphi(n\tau). Then the modular equation is solved in v instead and has the roots v=\varepsilon (n)\varphi(n\tau) and v=\varphi \tau+16m)/n :u=\varphi(n\tau) and :u=\varepsilon (n)\varphi\left(\frac\right) where \varepsilon (n) is 1 or −1 depending on whether 2 is a quadratic residue modulo ''n'' or not, respectively,Equivalently, \varepsilon (n)=(-1)^ (by the
law of quadratic reciprocity In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers. Due to its subtlety, it has many formulations, but the most standard st ...
).
and m\in\. For ''n'' = 5, we have the modular equation: :\Omega_5(u,v)=0\iff u^6 - v^6 + 5u^2v^2(u^2-v^2)+4uv(1-u^4v^4)=0 with six roots in u as shown above. The modular equation with ''n'' = 5 may be related to the Bring–Jerrard quintic by the following function of the six roots of the modular equation (In Hermite's ''Sur la théorie des équations modulaires et la résolution de l'équation du cinquième degré'', the first factor is incorrectly given as varphi(5\tau)+\varphi(\tau/5)/math>): :\Phi(\tau) = \left \varphi(5\tau) - \varphi\left(\frac\right)\rightleft varphi\left(\frac\right) - \varphi\left(\frac\right)\rightleft varphi\left(\frac\right) - \varphi\left(\frac\right)\right/math> Alternatively, the formula :\Phi (\tau)=2\sqrte^(1+e^-e^+e^-8e^-9e^+8e^-9e^+\cdots) is useful for numerical evaluation of \Phi (\tau). According to Hermite, the coefficient of e^ in the expansion is zero for every n\equiv 4\,(\operatorname5). The five quantities \Phi(\tau), \Phi(\tau+16), \Phi(\tau+32), \Phi(\tau+48), \Phi(\tau+64) are the roots of a quintic equation with coefficients rational in \varphi(\tau): :\Phi^5 - 2000\varphi^4(\tau)\psi^(\tau)\Phi - 64\sqrt\varphi^3(\tau)\psi^(\tau)\left + \varphi^8(\tau)\right= 0 which may be readily converted into the Bring–Jerrard form by the substitution: :\Phi = 2\sqrt varphi(\tau)\psi^4(\tau)x leading to the Bring–Jerrard quintic: :x^5 - x + a = 0 where :a = -\frac\quad (*) The Hermite–Kronecker–Brioschi method then amounts to finding a value for \tau that corresponds to the value of a, and then using that value of \tau to obtain the roots of the corresponding modular equation. We can use root finding algorithms to find \tau from the equation (*) (i.e. compute a
partial inverse In mathematics, the inverse function of a function (also called the inverse of ) is a function that undoes the operation of . The inverse of exists if and only if is bijective, and if it exists, is denoted by f^ . For a function f\colon X\ ...
of a). The roots of the Bring–Jerrard quintic are then given by: :x_r = \frac for r = 0, \ldots, 4. An alternative, "integral", approach is the following: Consider x^5-x+a=0 where a\in\mathbb\setminus\. Then :\tau=i\frac is a solution of :a=s\frac where :s=\begin-\operatorname\operatornamea&\text\operatornamea=0\\ \operatorname\operatornamea&\text\operatornamea\ne 0,\end :k^4 + A^2k^3 + 2k^2 - A^2k + 1 = 0,\quad (**) :A = \frac. The roots of the equation (**) are: :k = \tan \frac, \tan \frac, \tan \frac, \tan \frac where \sin \alpha = 4/A^2 (note that some important references erroneously give it as \sin \alpha = 1/(4A^2)). One of these roots may be used as the elliptic modulus k. The roots of the Bring–Jerrard quintic are then given by: :x_r = -s\frac for r = 0, \ldots, 4. It may be seen that this process uses a generalization of the
nth root In mathematics, a radicand, also known as an nth root, of a number ''x'' is a number ''r'' which, when raised to the power ''n'', yields ''x'': :r^n = x, where ''n'' is a positive integer, sometimes called the ''degree'' of the root. A root ...
, which may be expressed as: :\sqrt = \exp \left( \right) or more to the point, as :\sqrt = \exp \left(\frac\int^x_1\frac\right)=\exp\left(\frac\exp^ x\right). The Hermite–Kronecker–Brioschi method essentially replaces the exponential by an "elliptic transcendent", and the integral \int^x_1 dt/t (or the inverse of \exp on the real line) by an elliptic integral (or by a partial inverse of an "elliptic transcendent"). 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 In mathematics, Siegel modular forms are a major type of automorphic form. These generalize conventional ''elliptic'' modular forms which are closely related to elliptic curves. The complex manifolds constructed in the theory of Siegel modular form ...
s in place of the exponential/elliptic transcendents, and replaced the integral by a
hyperelliptic integral In mathematics, ''differential of the first kind'' is a traditional term used in the theories of Riemann surfaces (more generally, complex manifolds) and algebraic curves (more generally, algebraic varieties), for everywhere-regular differential 1 ...
.


Glasser's derivation

This derivation due to M. L. Glasser generalizes the series method presented earlier in this article to find a solution to any
trinomial In elementary algebra, a trinomial is a polynomial consisting of three terms or monomials. Examples of trinomial expressions # 3x + 5y + 8z with x, y, z variables # 3t + 9s^2 + 3y^3 with t, s, y variables # 3ts + 9t + 5s with t, s variables # a ...
equation of the form: : x^N - x + t=0 \,\! In particular, the quintic equation can be reduced to this form by the use of Tschirnhaus transformations as shown above. Let x = \zeta^\,, the general form becomes: : \zeta = e^ + t\phi(\zeta) \,\! where : \phi(\zeta) = \zeta^ \,\! A formula due to
Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangiaanalytic function In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex an ...
f \,, in the neighborhood of a root of the transformed general equation in terms of \zeta \,, above may be expressed as an
infinite series In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...
: : f(\zeta) = f(e^) + \sum^\infty_ \frac\frac \phi(a), ^n If we let f(\zeta) = \zeta^\, in this formula, we can come up with the root: : x_k = e^ - \frac\sum^\infty_\frac\cdot \frac : k=1,2, 3, \dots , N-1 \, By the use of the Gauss multiplication theorem the infinite series above may be broken up into a finite series of hypergeometric functions: :\psi_n(q) =\left(\frac\right)^q N^\frac =\left(\frac\right)^q N^\prod_^\frac : x_n = e^ - \frac\sqrt\sum^_\psi_n(q)_F_N \begin \frac, \ldots, \frac, 1; \\ pt \frac, \ldots, \frac, \frac; \\ pt \left(\frac\right)^N^N \end,\quad n=1,2, 3, \dots , N-1 : x_N = \sum_^ \frac\sqrt\sum^_\psi_m(q)_F_N \begin \frac, \ldots, \frac, 1; \\ pt \frac, \ldots, \frac, \frac; \\ pt \left(\frac\right)^N^N \end and the trinomial of the form has roots : _ \,\! :_ :_ :_ :_ 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: : \begin F_1(t) & = \,_4F_3\left(-\frac, \frac, \frac, \frac; \frac, \frac, \frac; \frac\right) \\ ptF_2(t) & = \,_4F_3\left(\frac, \frac, \frac, \frac; \frac, \frac, \frac; \frac\right) \\ ptF_3(t) & = \,_4F_3\left(\frac, \frac, \frac, \frac; \frac, \frac, \frac; \frac\right) \\ ptF_4(t) & = \,_4F_3\left(\frac, \frac, \frac, \frac; \frac, \frac, \frac; \frac\right) \end which are the hypergeometric functions that appear in the series formula above. The roots of the quintic are thus: : \begin x_1 & = & -tF_2(t) \\ ptx_2 & = & -F_1(t) & + & \fractF_2(t) & + & \fract^2F_3(t) & + & \fract^3F_4(t)\\ pt x_3 & = & F_1(t) & + & \fractF_2(t) & - & \fract^2F_3(t) & + & \fract^3F_4(t)\\ pt x_4 & = & -i F_1(t) & + & \fractF_2(t) & - & \fraci t^2F_3(t) & - & \fract^3F_4(t)\\ pt x_5 & = & i F_1(t) & + & \fractF_2(t) & + & \fraci t^2F_3(t) & - & \fract^3F_4(t) \end This is essentially the same result as that obtained by the following method.


The method of differential resolvents

James Cockle Sir James Cockle FRS FRAS FCPS (14 January 1819 – 27 January 1895) was an English lawyer and mathematician. Cockle was born on 14 January 1819. He was the second son of James Cockle, a surgeon, of Great Oakley, Essex. Educated at Charterho ...
and Robert Harley developed, in 1860, 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: :f(x) = x^5 - x + a\, and a function \,\phi(a)\, is to be determined such that: :f phi(a)= 0\, The function \,\phi\, must also satisfy the following four differential equations: : \begin \frac = 0 \\ pt\frac = 0 \\ pt\frac = 0 \\ pt\frac = 0 \end Expanding these and combining them together yields the differential resolvent: : \frac\frac - \frac\frac - \frac\frac - \frac\frac + \phi = 0 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, whose solution turns out to be identical to the series of hypergeometric functions that arose in Glasser's derivation above. This method may also be generalized to equations of arbitrarily high degree, with differential resolvents which are
partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function. The function is often thought of as an "unknown" to be sol ...
s, whose solutions involve hypergeometric functions of several variables. A general formula for differential resolvents of arbitrary univariate polynomials is given by Nahay's powersum formula.


Doyle–McMullen iteration

In 1989, Peter Doyle and Curt McMullen derived an iteration method that solves a quintic in Brioschi normal form: :x^5 - 10Cx^3 + 45C^2x - C^2 = 0.\, The iteration algorithm proceeds as follows: 1. Set Z = 1 - 1728C\, 2. Compute the rational function :: T_Z(w) = w - 12\frac\, :where g(Z,w)\, is a polynomial function given below, and g'\, is the
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...
of g(Z,w)\, with respect to w\, 3. Iterate T_Z _Z(w), on a random starting guess until it converges. Call the
limit point In mathematics, a limit point, accumulation point, or cluster point of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also contai ...
w_1\, and let w_2 = T_Z(w_1)\,. 4. Compute ::\mu_i = \frac\, :where h(Z,w)\, is a polynomial function given below. Do this for both w_1\, and w_2 = T_Z(w_1)\,. 5. Finally, compute ::x_i = \frac :for ''i'' = 1, 2. These are two of the roots of the Brioschi quintic. The two polynomial functions g(Z,w)\, and h(Z,w)\, are as follows: : \begin g(Z,w) = & 91125Z^6 \\ & + (-133650w^2 + 61560w - 193536)Z^5 \\ & + (-66825w^4 + 142560w^3 + 133056w^2 - 61140w + 102400)Z^4 \\ & + (5940w^6 + 4752w^5 + 63360w^4 - 140800w^3)Z^3 \\ & + (-1485w^8 + 3168w^7 - 10560w^6)Z^2 \\ & + (-66w^ + 440w^9)Z \\ & + w^ \\ pth(Z,w) = & (1215w - 648)Z^4 \\ & + (-540w^3 - 216w^2 - 1152w + 640)Z^3 \\ & + (378w^5 - 504w^4 + 960w^3)Z^2 \\ & + (36w^7 - 168w^6)Z \\ & + w^9 \end This iteration method produces two roots of the quintic. The remaining three roots can be obtained by using
synthetic division In algebra, synthetic division is a method for manually performing Euclidean division of polynomials, with less writing and fewer calculations than long division. It is mostly taught for division by linear monic polynomials (known as the Ruffini' ...
to divide the two roots out, producing a cubic equation. Due to the way the iteration is formulated, this method seems to always find two
complex conjugate In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
roots of the quintic even when all the quintic coefficients are real and the starting guess is real. This iteration method is derived from the symmetries of the
icosahedron In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes and . The plural can be either "icosahedra" () or "icosahedrons". There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrica ...
and is closely related to the method Felix Klein describes in his book.


See also

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Theory of equations In algebra, the theory of equations is the study of algebraic equations (also called "polynomial equations"), which are equations defined by a polynomial. The main problem of the theory of equations was to know when an algebraic equation has an ...


References


Notes


Other


Sources

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External links

* * * * {{DEFAULTSORT:Bring Radical Equations Polynomials Special functions