Bring radical

In algebra, a Bring radical or ultraradical is a root of the polynomial

$x^{5}+x+a,\,$

where a is a complex number. (The root is chosen so the radical of a real 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 radicals" section below.)

George Jerrard (1804–1863) showed that some quintic equations can be solved using radicals and Bring radicals, which had been introduced by Erland Bring (1736–1798). They can be used to obtain closed form solutions of quintic equations.

Normal Forms

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

$x^{5}+a_{4}x^{4}+a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}=0\,$

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:

$x^{5}+c_{2}x^{2}+c_{1}x+c_{0}=0\,$

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

$y_{k}=x_{k}^{2}+\alpha x_{k}+\beta \,$

the coefficients $\alpha$ and $\beta$ may be determined by using the resultant, or by means of the power-sum formulae of the quintic roots. This leads to a system of equations in $\alpha$ and $\beta$ 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:

$x^{5}+d_{1}x+d_{0}=0\,$

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.

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:

$z_{k}=x_{k}^{4}+\alpha x_{k}^{3}+\beta x_{k}^{2}+\gamma x_{k}+\delta$

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 package such as Mathematica or Maple. As might be expected, the resulting expressions can be enormous, taking many megabytes of storage for a general quintic with symbolic coefficients.

Regarded as an algebraic function, the solutions to

$x^{5}+d_{1}x+d_{0}=0\,$

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

$z={x \over (-d_{0}/5)^{1/5}}\,$

then we reduce the equation to the form

$x^{5}-5x-4t=0\,$

which involves x as an algebraic function of a single variable t. A similar transformation suffices to reduce the equation to

$x^{5}-x+a=0\,$

which is the form required by the Hermite-Kronecker-Brioschi method, Glasser's method, and the Cockle-Harley method of differential resolvents described below.

Bring radicals

As a function of the complex variable t, the roots x of

$x^{5}-5x-4t=0\,$

have branch points where the discriminant 800000(t⁴ − 1) is zero, which means at 1, −1, i and −i. Monodromy around any of the branch points exchanges two of the roots, leaving the rest fixed. For real values of t greater than or equal to −1, the largest real root is a function of t increasing monotonically from 1; we may call this function the Bring radical, BR(t). By taking a branch cut along the real axis from minus infinity to −1, we may extend the Bring radical to the entire complex plane, setting the value along the branch cut to be that obtained by analytically continuing around the upper half-plane.

More explicitly, let $a_{0}=3,a_{1}={1 \over 100},a_{2}=-{27 \over 400000},a_{3}={549/800000000}$, with subsequent a_i defined by the recurrence relationship

$a_{n+4}=-{\frac {185193}{5278000}}\,{\frac {2\,n+5}{n+4}}a_{n+3}$

$-{\frac {9747}{52780000}}\,{\frac {10\,{n}^{2}+40\,n+39}{\left(n+4\right)\left(n+3\right)}}a_{n+2}$

$-{\frac {57}{52780000}}\,{\frac {\left(2\,n+3\right)\left(10\,{n}^{2}+30\,n+17\right)}{\left(n+4\right)\left(n+3\right)\left(n+2\right)}}a_{n+1}$

$-{\frac {1}{6597500000}}\,{\frac {\left(5\,n+11\right)\left(5\,n+7\right)\left(5\,n+3\right)\left(5\,n-1\right)}{\left(n+4\right)\left(n+3\right)\left(n+2\right)\left(n+1\right)}}a_{n}.$ For complex values of t such that |t - 57| < 58, we then have

$\operatorname {BR} (t)=\sum _{n=0}^{\infty }a_{n}(t-57)^{n},\,$

which then can be analytically continued in the manner described.

The roots of x⁵ − 5x − 4t = 0 can now be expressed in terms of the Bring radical as

$r_{n}=i^{-n}\operatorname {BR} (i^{n}t)\,\!$

for n from 0 through 3, and

$r_{4}=-r_{0}-r_{1}-r_{2}-r_{3}\,\!$

for the fifth root.

Solution of the general quintic

We now may express the roots of any polynomial

$x^{5}+px+q\,$

in terms of the Bring radical as

$\left(-{\frac {p}{5}}\right)^{\frac {1}{4}}\operatorname {BR} \left({\frac {(-5/p)^{\frac {5}{4}}q}{-4}}\right)$

and its four conjugates. 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.

Other Characterizations

Many other characterizations of the Bring radical have been developed, the first of which in terms of elliptic 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:

$x^{5}-x+a=0\,$

into which any quintic equation may be reduced by means of Tschirnhaus transformations as has been shown. He observed that elliptic functions had an analogous role to play in the solution of the Bring-Jerrard quintic as the trigonometric functions had for the cubic. If $K$ and $K'$ are the periods of an elliptic integral of the first kind:

$K=\int _{0}^{\frac {\pi }{2}}{\frac {d\varphi }{\sqrt {1-k^{2}\sin ^{2}\varphi }}}$

$K'=\int _{0}^{\frac {\pi }{2}}{\frac {d\varphi }{\sqrt {1-k'^{2}\sin ^{2}\varphi }}}$

the elliptic nome is given by:

$q=\exp(-\pi K'/K)\,$

and

$k^{2}+k'^{2}=1\,$

With

$q=\exp(-\pi K'/K)=\exp(i\pi \tau )\,$

define the two elliptic modular functions:

${\sqrt[{4}]{k}}=\varphi (\tau )={\sqrt {\vartheta _{10}(0;\tau )/\vartheta _{00}(0;\tau )}}$

${\sqrt[{4}]{k'}}=\psi (\tau )={\sqrt {\vartheta _{01}(0;\tau )/\vartheta _{00}(0;\tau )}}$

where $\vartheta _{00}(0;\tau )$ and similar are Jacobi theta functions.

If $n$ is a prime number, we can define two values $u$ and $v$ as follows:

$v=\varphi (n\tau )\,$

and

$u=\varphi (\tau )\,$

The parameters $u$ and $v$ are linked by an equation of degree $n+1$ known as the modular equation, whose $n+1$ roots are given by:

$\epsilon \varphi (n\tau )\,$

and

$\varphi \left({\frac {\tau +16m}{n}}\right)\,$

where $\epsilon$ 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:

$u^{6}-v^{6}+5u^{2}v^{2}(u^{2}-v^{2})+4uv(1-u^{4}v^{4})=0\,$

with six roots as shown above.

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:

$\Phi (\tau )=(\varphi (5\tau )+\varphi (\tau /5))(\varphi ((\tau +16)/5)-\varphi ((\tau +64)/5))(\varphi ((\tau +32)/5)-\varphi ((\tau +48)/5))\,$

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 $\varphi (\tau )$:

$\Phi ^{5}-2000\varphi ^{4}(\tau )\psi ^{16}(\tau )\Phi -1600{\sqrt {5}}\varphi ^{3}(\tau )\psi ^{16}(\tau )(1+\varphi ^{8}(\tau ))=0\,$

which may be readily converted into the Bring-Jerrard form by the substitution:

$\Phi ={\sqrt[{4}]{2^{4}5^{3}}}\varphi (\tau )\psi ^{4}(\tau )x\,$

leading to the Bring-Jerrard quintic:

$x^{5}-x+a=0\,$

where

$a={\frac {2(1+\varphi ^{8}(\tau ))}{{\sqrt[{4}]{5^{5}}}\varphi ^{2}(\tau )\psi ^{4}(\tau )}}\,$

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. To do this, let

$A={\frac {a{\sqrt[{4}]{5^{5}}}}{2}}$

and calculate the required elliptic modulus $k$ by solving the quartic equation:

$k^{4}+A^{2}k^{3}+2k^{2}-A^{2}k+1=0\,$

The roots of this equation are:

$k=\tan(\alpha /4),\tan((\alpha +2\pi )/4),\tan((\pi -\alpha )/4),\tan(((3\pi -\alpha )/4)\,$

where $\sin(\alpha )=1/(4A^{2})$. Any of these roots may be used as the elliptic modulus for the purposes of the method. The value of $\tau$ may be easily obtained from the elliptic modulus $k$ by the relations given above. The roots of the Bring-Jerrard quintic are then given by:

$x_{i}={\frac {\Phi (\tau +16i)}{{\sqrt[{4}]{2^{4}5^{3}}}\varphi (\tau )\psi ^{4}(\tau )}}$

where $i=0,\ldots ,4$.

It may be seen that this process uses a generalization of the nth root, which may be expressed as:

${\sqrt[{n}]{x}}=\exp \left({\frac {1}{n}}\ln x\right)$

or more to the point, as

${\sqrt[{n}]{x}}=\exp \left({\frac {1}{n}}\int _{1}^{x}{\frac {dt}{t}}\right).$

The Hermite-Kronecker-Brioschi method essentially replaces the exponential by an elliptic modular function, and the integral $\int _{1}^{a}{\frac {dt}{t}}$ 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 forms in place of the exponential/elliptic modular function, and the integral by a hyperelliptic integral.

Glasser's derivation

This derivation due to M. L. Glasser finds a solution to any trinomial equation of the form:

$x^{N}-x+t\,\!$

In particular, the quintic equation can be reduced to this form by the use of Tschirnhaus transformations as shown above. Let $x=\zeta ^{-1/(N-1)}$, the general form becomes:

$\zeta =e^{2\pi i}+t\phi (\zeta )\,\!$

where

$\phi (\zeta )=\zeta ^{N/(N-1)}\,\!$

A formula due to Lagrange states that for any analytic function f in the neighborhood of a root of the transformed general equation in terms of ζ above may be expressed as an infinite series:

$f(\zeta )=f(e^{2\pi i})+\sum _{n=1}^{\infty }{\frac {t^{n}}{n!}}{\frac {d^{n-1}}{da^{n-1}}}[f'(a)|\phi (a)|^{n}]_{a=e^{2\pi i}}$

If we let $f(\zeta )=\zeta ^{-1/(N-1)}$ in this formula, we can come up with the root:

$x_{1}=\exp(-2\pi i/(N-1))-{\frac {t}{N-1}}\sum _{n=0}^{\infty }{\frac {(te^{2\pi i/(N-1)})^{n}}{\Gamma (n+2)}}{\frac {\Gamma ({\frac {Nn}{N-1}}+1)}{\Gamma ({\frac {n}{N-1}}+1)}}$

A further N−2 roots may be found by replacing $\exp(-2\pi i/(N-1))$ by the other (N−1)th roots of unity, and the last root by using any of the symmetric function relations between the roots of a polynomial (e.g. the sum of all the roots of any polynomial in the trinomial form above —N greater than 2— must be 0). By the use of the Gauss multiplication theorem the infinite series above may be broken up into a finite series of hypergeometric functions:

$\psi (q)=({\frac {\omega t}{N-1}})^{q}n^{qN/(N-1)}{\frac {\prod _{k=0}^{N-1}\Gamma ({\frac {Nq/(N-1)+1+k}{N}})}{\Gamma ({\frac {q}{N-1}}+1)\prod _{k=0}^{N-2}\Gamma ({\frac {q+k+2}{N-1}})}}$

$x_{1}=\omega ^{-1}-{\frac {t}{(N-1)^{2}}}{\sqrt {\frac {N}{2\pi (N-1)}}}\sum _{q=0}^{N-2}\psi (q)_{N+1}F_{N}{\begin{bmatrix}{\frac {qN/(N-1)+1}{N}},\ldots ,{\frac {qN/(N-1)+N}{N}},1;\\{\frac {q+2}{N-1}},\ldots ,{\frac {q+N}{N-1}},{\frac {q}{N-1}}+1;\\({\frac {t\omega }{N-1}})^{N-1}N^{N})\end{bmatrix}}$

where $\omega =\exp(2\pi i/(N-1))$. 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:

${\begin{matrix}F_{1}(t)&=&F_{2}(t)\\F_{2}(t)&=&\,_{4}F_{3}(1/5,&2/5,&3/5,&4/5;&1/2,&3/4,&5/4;&3125t^{4}/256)\\F_{3}(t)&=&\,_{4}F_{3}(9/20,&13/20,&17/20,&21/20;&3/4,&5/4,&3/2;&3125t^{4}/256)\\F_{4}(t)&=&\,_{4}F_{3}(7/10,&9/10,&11/10,&13/10;&5/4,&3/2,&7/4;&3125t^{4}/256)\end{matrix}}$

which are the hypergeometric functions which appear in the series formula above. The roots of the quintic are thus:

${\begin{matrix}x_{1}&=&tF_{1}(t)\\x_{2}&=&-F_{1}(t)&+{\frac {1}{4}}tF_{2}(t)&+{\frac {5}{32}}t^{2}F_{3}(t)&+{\frac {5}{32}}t^{3}F_{3}(t)\\x_{3}&=&-F_{1}(t)&+{\frac {1}{4}}tF_{2}(t)&-{\frac {5}{32}}t^{2}F_{3}(t)&+{\frac {5}{32}}t^{3}F_{3}(t)\\x_{4}&=&-iF_{1}(t)&+{\frac {1}{4}}tF_{2}(t)&-{\frac {5}{32}}it^{2}F_{3}(t)&-{\frac {5}{32}}t^{3}F_{3}(t)\\x_{5}&=&iF_{1}(t)&+{\frac {1}{4}}tF_{2}(t)&+{\frac {5}{32}}it^{2}F_{3}(t)&-{\frac {5}{32}}t^{3}F_{3}(t)\\\end{matrix}}$

This is essentially the same result as that obtained by the method of differential resolvents developed by James Cockle and Robert Harley in 1860.

The Method of differential resolvents

James Cockle and Robert Harley 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:

$f(x)=x^{5}-x+a$

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:

${\frac {df(\phi (a))}{da}}=0$

${\frac {d^{2}f(\phi (a))}{da^{2}}}=0$

${\frac {d^{3}f(\phi (a))}{da^{3}}}=0$

${\frac {d^{4}f(\phi (a))}{da^{4}}}=0$

Expanding these and combining them together yields the differential resolvent:

${\frac {(256-3125a^{4})}{1155}}{\frac {d^{4}\phi }{da^{4}}}-{\frac {6250a^{3}}{231}}{\frac {d^{3}\phi }{da^{3}}}-{\frac {4875a^{2}}{77}}{\frac {d^{2}\phi }{da^{2}}}-{\frac {2125a}{77}}{\frac {d\phi }{da}}+\phi =0$

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 equations, whose solutions involve hypergeometric functions of several variables.

See also

• Theory of equations

References

• Felix Klein, 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

External links

• M. Hazewinkel (2001) [1994], "Tschirnhausen transformation", Encyclopedia of Mathematics, EMS Press {{citation}}: Unknown parameter |urlname= ignored (help)
• Weisstein, Eric W. "Bring-Jerrard Quintic Form". MathWorld.
• Weisstein, Eric W. "Bring Quintic Form". MathWorld.