Bring radical

Normal forms

The quintic equation is rather difficult to obtain solutions for directly, with five independent coefficients in its most general form:

x5+a4x4+a3x3+a2x2+a1x+a0=0.

The various methods for solving the quintic that have been developed generally attempt to simplify the quintic using Tschirnhaus transformations 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:

y5+c2y2+c1y+c0=0

If the roots of a general quintic and a principal quintic are related by a quadratic Tschirnhaus transformation

yk=xk2+αxk+β,

the coefficients α and β may be determined by using the resultant, or by means of the power sums of the roots and Newton's identities. 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'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:

v5+d1v+d0=0.

Using the power-sum formulae again with a cubic transformation as Tschirnhaus 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:

vk=yk4+αyk3+βyk2+γyk+δ.

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 package such as Mathematica

or Maple.

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

v5+d1v+d0=0

involve two variables, d1 and d0; 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=v−d14

Brioschi normal form

There is another one-parameter normal form for the quintic equation, known as Brioschi normal form

w5−10Cw3+45C2w−C2=0,

which can be derived by using the rational Tschirnhaus transformation

wk=λ+μxkxk2C−3

to relate the roots of a general quintic to a Brioschi quintic. The values of the parameters λ and μ may be derived by using polyhedral functions on the Riemann sphere, and are related to the partition of an object of icosahedral symmetry into five objects of tetrahedral symmetry.

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 for Bring radicals, as well as a representation in terms of hypergeometric functions can be derived as follows. The equation x5+x+a=0 can be rewritten as x5+x=−a. By setting f(x)=x5+x, the desired solution is x=f−1(−a)=−f−1(a) since f(x) is odd.

The series for f−1 can then be obtained by reversion of the Taylor series for f(x) (which is simply x+x5), giving

BR⁡(a)=−f−1(a)=∑k=0∞(5kk)(−1)k+1a4k+14k+1=−a+a5−5a9+35a13−285a17+⋯,

where the absolute values of the coefficients form sequence A002294 in the OEIS. The radius of convergence of the series is 4/(5⋅54)≈0.53499.

In hypergeometric form, the Bring radical can be written as

BR⁡(a)=−a4F3(15,25,35,45;12,34,54;−5(5a4)4).

Solution of the general quintic

The roots of the polynomial

x5+px+q

can be expressed in terms of the Bring radical as

p4BR⁡(p−54q)

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 and modular functions) by Charles Hermite 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

and Leopold Kronecker

came upon equivalent solutions. Hermite arrived at this solution by generalizing the well-known solution to the cubic equation in terms of trigonometric functions and finds the solution to a quintic in Bring–Jerrard form:

x5−x+a=0

into which any quintic equation may be reduced by means of Tschirnhaus transformations as has been shown. He observed that elliptic functions 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 them as the complete elliptic integrals of the first kind:

K(k)=∫0π2dφ1−k2sin2φ

K′(k)=∫0π2dφ1−k'2sin2φ

where

k2+k'2=1.

Define the two "elliptic transcendents":

φ(τ)=∏j=1∞tanh⁡(2j−1)πi2τ=2eπiτ/8∏j=1∞1+e2jπiτ1+e(2j−1)πiτ,Im⁡τ>0

ψ(τ)=∏j=1∞tanh⁡(2j−1)πτ2i,Im⁡τ>0

They can be equivalently defined by infinite series:

φ(τ)=2eπiτ/8∑j∈ℤe(2j2+j)πiτ∑j∈ℤej2πiτ=2eπiτ/8(1−eπiτ+2e2πiτ−3e3πiτ+4e4πiτ−6e5πiτ+9e6πiτ−⋯),Im⁡τ>0ψ(τ)=∑j∈ℤ(−1)je2j2πiτ∑j∈ℤej2πiτ=1−2eπiτ+2e2πiτ−4e3πiτ+6e4πiτ−8e5πiτ+12e6πiτ−⋯,Im⁡τ>0

The method of differential resolvents

James Cockle

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)=x5−x+a

and a function ϕ(a) is to be determined such that:

f[ϕ(a)]=0

The function ϕ must also satisfy the following four differential equations:

df[ϕ(a)]da=0d2f[ϕ(a)]da2=0d3f[ϕ(a)]da3=0d4f[ϕ(a)]da4=0

Expanding these and combining them together yields the differential resolvent:

(256−3125a4)1155d4ϕda4−6250a3231d3ϕda3−4875a277d2ϕda2−2125a77dϕda+ϕ=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.

This method may also be generalized to equations of arbitrarily high degree, with differential resolvents which are partial differential equations, 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:

x5−10Cx3+45C2x−C2=0.

The iteration algorithm proceeds as follows:

Set Z=1−1728C

Compute the rational function TZ(w)=w−12g(Z,w)g′(Z,w) where g(Z,w) is a polynomial function given below, and g′ is the derivative of g(Z,w) with respect to w

Iterate TZ[TZ(w)] on a random starting guess until it converges. Call the limit point w1 and let w2=TZ(w1).

Compute μi=100Z(Z−1)h(Z,wi)g(Z,wi) where h(Z,w) is a polynomial function given below. Do this for both w1 and w2=TZ(w1).

Finally, compute xi=(9+15i)μi+(9−15i)μ3−i90 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:

g(Z,w)=91125Z6+(−133650w2+61560w−193536)Z5+(−66825w4+142560w3+133056w2−61140w+102400)Z4+(5940w6+4752w5+63360w4−140800w3)Z3+(−1485w8+3168w7−10560w6)Z2+(−66w10+440w9)Z+w12h(Z,w)=(1215w−648)Z4+(−540w3−216w2−1152w+640)Z3+(378w5−504w4+960w3)Z2+(36w7−168w6)Z−w9

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. 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 from the symmetries of the icosahedron and is closely related to the method Felix Klein describes in his book.