Draft:Adamchik transformation

The mathematical Adamchik Transformation is a type of mapping on Principal Quintic polynomials with the aim of transforming these polynomials into Bring Jerrard Quintic polynomials. This transformation is a special case of the Tschirnhaus Transformation and it uses a quartic transformation key. The Adamchik Transformation is developed by the mathematicians Victor Adamchik and David Jeffrey from the University of Southern California in the first years of the third millennium. And exactly this polynomial transformation is described in the essay Polynomial Transformations of Tschirnhaus, Bring and Jerrard by Adamchik and Jeffrey, and it was published in 2003.

Definition

The Adamchik Transformation transforms a Principal Quintic polynomial equation into a Bring Jerrard equation by using a quartic key, a key of fourth degree. The Principal Quintic is a special equation of fifth degree that contains a quintic as well as a quadratic, a linear and an absolute term but does neither contain a quartic nor a cubic term. Thus it fulfills the following pattern:

$x^{5}+rx^{2}+sx+t=0$

This Principal equation of fifth degree shall be transformed into a Bring Jerrard equation after this pattern:

$(x^{4}+ax^{3}+bx^{2}+cx+d)^{5}+m(x^{4}+ax^{3}+bx^{2}+cx+d)+n=0$

Along with the Polynomial Division in partial pattern, this accurately means that the left hand side of the under equation in this section must be able to be divided by the left hand side of the upper equation without any polynomial rest. By using this transformation the quadratic term of the Quintic system shall be eliminated. In this way the Bring Jerrard Quintic form appears.

The Bring Jerrard equation is a special fifth degree equation that does only contain a quintic, a linear and an absolute term. Along with the Abel Ruffini Theorem, the regular case of Bring Jerrard equations can not be solved in an elementary way. But that equation can be easily solved with the modular Elliptic Jacobi Theta Function and by using the elliptic modulus k in Legendre form and computing its Elliptic Nome function value. The determination of the elliptic modulus k for the corresponding Bring Jerrard equation is researched by Charles Hermite and Francesco Brioschi and published in the essay Sur la résolution de l'Équation du cinquiéme degré Comptes rendus from 1858.

Determination of the coefficients

Basic pattern

Given is the mentioned pattern of the Adamchik Transformation in its parametrized form:

$x^{5}+rx^{2}+sx+t=0$

$y=x^{4}+ax^{3}+bx^{2}+cx+d$

$y^{5}+my+n=0$

To fulfill the mentioned polynomial division without any polynomial rest, and in the same way to create a Bring Jerrard that solves y and afterwards leads to the x solution, it is necessary to find out the relations between the coefficients of the quartic key and the coefficients of the principal quintic basic equation. The mathematicians Victor Adamchik and David Jeffrey worked out the exact equations that lead to the coefficients of the quartic key and of the Bring Jerrard final form.

Using a quartic key for that Tschirnhaus Transformation in the shape of the Adamchik Transformation is relevant. Only in a transformation key of fourth degree the coefficients of the key can stand in an elementary relation to the coefficients of the Principal Quintic. In the case of using a cubic key the resulting coefficients of that key require the solving of an equation of sixth degree. And regular equations of sixth degree have no elementary solution. But by using the quartic key this problem can be deactivated and there are coefficient combinations in elementary relation to the coefficients of the Principal equation.

Transformation patterns of the equations

If the solution of the given Principal Quintic equation is entered into another polynomial, the corresponding quintic equation for this polynomial contains other coefficients that stand in rational polynomial relation to the coefficients of the Principal form. In the following, an algorithm that leads to the new coefficients, shall be shown:

First of all the quartic key itself shall be solved indirectly by combining the x solution and the other three solutions of the same quartic key:

$y=x^{4}+ax^{3}+bx^{2}+cx+d$

This can be made with the Theorem of Vieta in the following way:

$\mathrm {I} )\,\,\,u+v+w+x=-a$

$\mathrm {II} )\,\,\,uv+uw+ux+vw+vx+wx=b$

$\mathrm {III} )\,\,\,uvw+uvx+uwx+vwx=-c$

$\mathrm {IV} )\,\,\,uvwx=d-y$

In the next step the Principal Quintic equation is multiplied with the same equation pattern in the name of the other three solutions of the quartic key:

$(u^{5}+ru^{2}+su+t)(v^{5}+rv^{2}+sv+t)(w^{5}+rw^{2}+sw+t)(x^{5}+rx^{2}+sx+t)=0$

This expression leads to the Tschirnhaus transformed new quintic equation in relation to y and afterwards to all relations of the coefficients. By putting in the four clues of the equations marked with Roman numbers into the last mentioned product of the four Principal polynomials this transformed new equation in relation to y appears:

$0=(u^{5}+ru^{2}+su+t)(v^{5}+rv^{2}+sv+t)(w^{5}+rw^{2}+sw+t)(x^{5}+rx^{2}+sx+t)=$

$\,=-\,y^{5}+(-4s-3ar+5d)\,y^{4}+$

+ (4*r*t - 6*s^2 - 5*a*r*s - 3*a^2*r^2 + 3*b*r^2 - 5*a*b*t - 5*c*t + 16*d*s - 2*b^2*s - 4*a*c*s + 12*a*d*r - 3*b*c*r - 10*d^2)*y^3 +

+ (r^4 - 4*s^3 + 8*r*s*t + a*r^2*t - a*r*s^2 + 5*a^2*t^2 + 5*b*t^2 - a^2*r^2*s + 2*b*r^2*s + 3*a^3*s*t - 2*a*b*s*t - 11*c*s*t - a^3*r^3 + 3*a*b*r^3 - 3*c*r^3 - 7*a^2*b*r*t + a*c*r*t + 8*b^2*r*t - 12*d*r*t + 4*a^2*b*s^2 - 8*a*c*s^2 - 4*b^2*s^2 + 18*d*s^2 - 5*a^2*c*r*s + a*b^2*r*s + 15*a*d*r*s + 2*b*c*r*s + 9*a^2*d*r^2 - 3*a*b*c*r^2 + b^3*r^2 - 9*b*d*r^2 + 3*c^2*r^2 + 15*a*b*d*t - 5*a*c^2*t - 5*b^2*c*t + 15*c*d*t + 12*a*c*d*s + 6*b^2*d*s - 4*b*c^2*s - 24*d^2*s - 18*a*d^2*r + 9*b*c*d*r - c^3*r + 10*d^3)*y^2 +

+ (-2*r^2*t^2 + 4*r*s^2*t - s^4 - 3*a*r^2*s*t + a*r*s^3 - 5*a*t^3 - a^2*s*t^2 + 2*b*r^3*t - b*r^2*s^2 + 6*b*s*t^2 + 2*a^3*r*t^2 - 4*a*b*r*t^2 + 7*c*r*t^2 + -a^3*s^2*t + 3*a*b*s^2*t + c*r^3*s - 7*c*s^2*t + 3*a^4*r*s*t - 11*a^2*b*r*s*t + 10*a*c*r*s*t + 4*b^2*r*s*t - 16*d*r*s*t - a^4*s^3 + 4*a^2*b*s^3 - 4*a*c*s^3 - 2*b^2*s^3 - 2*d*r^4 + 8*d*s^3 - 2*a^3*b*r^2*t + 6*a*b^2*r^2*t - 2*a*d*r^2*t - 6*b*c*r^2*t + a^3*b*r*s^2 - a^2*c*r*s^2 - 3*a*b^2*r*s^2 + 2*a*d*r*s^2 + 5*b*c*r*s^2 + 5*a^3*c*t^2 - 5*a^2*b^2*t^2 - 10*a^2*d*t^2 + 5*a*b*c*t^2 + 5*b^3*t^2 - 10*b*d*t^2 - 5*c^2*t^2 - a^3*c*r^2*s + 2*a^2*d*r^2*s + 3*a*b*c*r^2*s - 4*b*d*r^2*s - 3*c^2*r^2*s - 6*a^3*d*s*t + 7*a^2*b*c*s*t - a*b^3*s*t + 4*a*b*d*s*t - 13*a*c^2*s*t - 3*b^2*c*s*t + 22*c*d*s*t + 2*a^3*d*r^3 - 6*a*b*d*r^3 + 6*c*d*r^3 + 14*a^2*b*d*r*t - 3*a^2*c^2*r*t - 6*a*b^2*c*r*t - 2*a*c*d*r*t + 2*b^4*r*t - 16*b^2*d*r*t + 9*b*c^2*r*t + 12*d^2*r*t - 8*a^2*b*d*s^2 - 2*a^2*c^2*s^2 + 4*a*b^2*c*s^2 + 16*a*c*d*s^2 - b^4*s^2 + 8*b^2*d*s^2 - 4*b*c^2*s^2 - 18*d^2*s^2 + 10*a^2*c*d*r*s - 2*a*b^2*d*r*s - 3*a*b*c^2*r*s - 15*a*d^2*r*s + b^3*c*r*s - 4*b*c*d*r*s + 3*c^3*r*s - 9*a^2*d^2*r^2 + 6*a*b*c*d*r^2 - 2*b^3*d*r^2 + 9*b*d^2*r^2 - 6*c^2*d*r^2 - 15*a*b*d^2*t + 10*a*c^2*d*t + 10*b^2*c*d*t - 5*b*c^3*t - 15*c*d^2*t - 12*a*c*d^2*s - 6*b^2*d^2*s + 8*b*c^2*d*s - c^4*s + 16*d^3*s + 12*a*d^3*r - 9*b*c*d^2*r + 2*c^3*d*r - 5*d^4)*y -

- c s^3 t - a b r s t^2 + 3 c r s t^2 + b s^2 t^2 + a^2 r t^3 - 2 b r t^3 - a s t^3 + t^4 - 2*a*c*r^2*t^2 + a*c*r*s^2*t + b^2*r^2*t^2 + 2*d*r^2*t^2 - 4*d*r*s^2*t + d*s^4 - a^5*t^3 + 5*a^3*b*t^3 - 5*a^2*c*t^3 - 5*a*b^2*t^3 + 3*a*d*r^2*s*t - a*d*r*s^3 + 5*a*d*t^3 - b*c*r^2*s*t + 5*b*c*t^3 + a^4*b*s*t^2 - a^3*c*s*t^2 - 4*a^2*b^2*s*t^2 + a^2*d*s*t^2 + 7*a*b*c*s*t^2 + 2*b^3*s*t^2 - 2*b*d*r^3*t + b*d*r^2*s^2 - 6*b*d*s*t^2 + c^2*r^3*t - 3*c^2*s*t^2 + 2*a^4*c*r*t^2 - a^3*b^2*r*t^2 - 3*a^3*d*r*t^2 - 6*a^2*b*c*r*t^2 + 3*a*b^3*r*t^2 + 4*a*b*d*r*t^2 + 7*a*c^2*r*t^2 - 3*b^2*c*r*t^2 - 7*c*d*r*t^2 - a^4*c*s^2*t + a^3*d*s^2*t + 4*a^2*b*c*s^2*t - 3*a*b*d*s^2*t - 4*a*c^2*s^2*t - 2*b^2*c*s^2*t - c*d*r^3*s + 7*c*d*s^2*t - 3*a^4*d*r*s*t + a^3*b*c*r*s*t + 11*a^2*b*d*r*s*t - a^2*c^2*r*s*t - 3*a*b^2*c*r*s*t - 10*c*a*d*r*s*t - 4*b^2*d*r*s*t + 5*b*c^2*r*s*t + 8*d^2*r*s*t + a^4*d*s^3 - 4*a^2*b*d*s^3 + 4*a*c*d*s^3 + 2*b^2*d*s^3 + d^2*r^4 - 4*d^2*s^3 + 2*a^3*b*d*r^2*t - a^3*c^2*r^2*t - 6*a*b^2*d*r^2*t + 3*a*b*c^2*r^2*t + a*d^2*r^2*t + 6*b*c*d*r^2*t - 3*c^3*r^2*t - a^3*b*d*r*s^2 + a^2*c*d*r*s^2 + 3*a*b^2*d*r*s^2 - a*d^2*r*s^2 - 5*b*c*d*r*s^2 - 5*a^3*c*d*t^2 + 5*a^2*b^2*d*t^2 + 5*a^2*b*c^2*t^2 + 5*a^2*d^2*t^2 - 5*a*b^3*c*t^2 - 5*a*b*c*d*t^2 - 5*a*c^3*t^2 + b^5*t^2 - 5*b^3*d*t^2 + 5*b^2*c^2*t^2 + 5*b*d^2*t^2 + 5*c^2*d*t^2 + a^3*c*d*r^2*s - a^2*d^2*r^2*s - 3*a*b*c*d*r^2*s + 2*b*d^2*r^2*s + 3*c^2*d*r^2*s + 3*a^3*d^2*s*t - 7*a^2*b*c*d*s*t - 2*a^2*c^3*s*t + a*b^3*d*s*t + 4*a*b^2*c^2*s*t - 2*a*b*d^2*s*t + 13*a*c^2*d*s*t - b^4*c*s*t + 3*b^2*c*d*s*t - 4*b*c^3*s*t - 11*c*d^2*s*t - a^3*d^2*r^3 + 3*a*b*d^2*r^3 - 3*c*d^2*r^3 - 7*a^2*b*d^2*r*t + 3*a^2*c^2*d*r*t + 6*a*b^2*c*d*r*t - 3*a*b*c^3*r*t + a*c*d^2*r*t - 2*b^4*d*r*t + b^3*c^2*r*t + 8*b^2*d^2*r*t - 9*b*c^2*d*r*t + 3*c^4*r*t - 4*d^3*r*t + 4*a^2*b*d^2*s^2 + 2*a^2*c^2*d*s^2 - 4*a*b^2*c*d*s^2 - 8*a*c*d^2*s^2 + b^4*d*s^2 - 4*b^2*d^2*s^2 + 4*b*c^2*d*s^2 + 6*d^3*s^2 - 5*a^2*c*d^2*r*s + a*b^2*d^2*r*s + 3*a*b*c^2*d*r*s + 5*a*d^3*r*s - b^3*c*d*r*s + 2*b*c*d^2*r*s - 3*c^3*d*r*s + 3*a^2*d^3*r^2 - 3*a*b*c*d^2*r^2 + b^3*d^2*r^2 - 3*b*d^3*r^2 + 3*c^2*d^2*r^2 + 5*a*b*d^3*t - 5*a*c^2*d^2*t - 5*b^2*c*d^2*t + 5*b*c^3*d*t - c^5*t + 5*c*d^3*t + 4*a*c*d^3*s + 2*b^2*d^3*s - 4*b*c^2*d^2*s + c^4*d*s - 4*d^4*s - 3*a*d^4*r + 3*b*c*d^3*r - c^3*d^2*r + d^5

$\mathrm {Summandized\,\,Form}$

Exactly this summandized form directly leads to the clues that connect the coefficients of the Principal form with the coefficient of the quartic key and also with the coefficients of the Bring Jerrard form that is strived for. By setting the summandization negative the coefficients of the Quintic result can be read in its complete form.

Clues of the Transformation

In the now following step the shown summandized quintic form must be set equal with the Bring Jerrard Form. The coefficients of the quartic, the cubic and the quadratic term must be set to zero. Therefore these are the first two clues for the determination of the coefficients of the quartic key:

Equation for the quartic term of the resulting quintic:

$\mathrm {I} )\,\,\,4s+3ar-5d=0$

Equation for the cubic term of the resulting quintic:

$\mathrm {II} )\,\,\,-4rt+6s^{2}+5ars+3a^{2}r^{2}-3br^{2}+5abt+5ct-16ds+2b^{2}s+4acs-12adr+3bcr+10d^{2}=0$

Equation for the quadratic term of the resulting quintic:

$\mathrm {III} )$ r*c^3 + (-3*r^2 + 5*a*t + 4*b*s)*c^2 + (11*s*t + 3*r^3 - a*r*t + 8*a*s^2 + 5*a^2*r*s - 2*b*r*s + 3*a*b*r^2 + 5*b^2*t - 15*d*t - 12*a*d*s - 9*b*d*r)*c - (r^4 - 4*s^3 + 8*r*s*t + a*r^2*t - a*r*s^2 + 5*a^2*t^2 + 5*b*t^2 - a^2*r^2*s + 2*b*r^2*s + 3*a^3*s*t - 2*a*b*s*t - a^3*r^3 + 3*a*b*r^3 - 7*a^2*b*r*t + 8*b^2*r*t - 12*d*r*t + 4*a^2*b*s^2 - 4*b^2*s^2 + 18*d*s^2 + a*b^2*r*s + 15*a*d*r*s + 9*a^2*d*r^2 + b^3*r^2 - 9*b*d*r^2 + 15*a*b*d*t + 6*b^2*d*s - 24*d^2*s - 18*a*d^2*r + 10*d^3) = 0

This equation system of three equations but four unknowns a, b, c, d has infinite many solutions for those four unknowns. But only a few solutions of that equation stand in an elementary relation to the coefficients r and s and t of the given Principal Quintic. However one solution of that system of three equation clues does always have an elementary relation to the coefficients of the Principal Quintic and can be generated by adding a fourth condition. In the coefficient of the cubic term of the final form there is only one unknown variable that appears linear, and that is variable c accurately. And this variable, the unknown c appears in the coefficient of the final quadratic term in a cubic equation. So it is useful to take the independence of the cubic term from the variable c as the further condition. To make the coefficient of the cubic term of the final equation independent in this way, that cubic coefficient term in relation to c must have a horizontal slope. Therefore the slope must be set equal to zero. This looks exactly like that:

$\mathrm {II} )\,\,\,-4rt+6s^{2}+5ars+3a^{2}r^{2}-3br^{2}+5abt-16ds+2b^{2}s-12adr+10d^{2}+(5t+4as+3br)c=0$

New condition:

$5t+4as+3br=0$

The expression in the round brackets is set equal to zero because this is the mentioned slope. So the further condition is formulated. In this way a new equation system can be built for the unknowns a, b and d that is succeeded by the already mentioned equation from the quadratic term for solving the left unknown c too. The determination of c requires the solving of a cubic equation. And so the quadratic key for the Tschirnhaus Transformation is synthesized.

Synthesis of the Transformation

Synthesis of the quartic key

Along with the already given explanation a system of three equations and three unknowns can be created. By adding the further demanded condition and simplifying the equation system, the following system for the determination of a, b and d is generated:

$\mathrm {I^{*}} )\,\,\,4s+3ar-5d=0$

$\mathrm {II^{*}} )\,\,\,5t+4as+3br=0$

$\mathrm {III^{*}} )\,\,\,3rt-2s^{2}+15abt+6b^{2}s-3adr+ds=0$

In a successive way, the unknown c can be solved by the mentioned cubic equation:

$\mathrm {IV^{*}} )$ r*c^3 + (-3*r^2 + 5*a*t + 4*b*s)*c^2 + (11*s*t + 3*r^3 - a*r*t + 8*a*s^2 + 5*a^2*r*s - 2*b*r*s + 3*a*b*r^2 + 5*b^2*t - 15*d*t - 12*a*d*s - 9*b*d*r)*c - (r^4 - 4*s^3 + 8*r*s*t + a*r^2*t - a*r*s^2 + 5*a^2*t^2 + 5*b*t^2 - a^2*r^2*s + 2*b*r^2*s + 3*a^3*s*t - 2*a*b*s*t - a^3*r^3 + 3*a*b*r^3 - 7*a^2*b*r*t + 8*b^2*r*t - 12*d*r*t + 4*a^2*b*s^2 - 4*b^2*s^2 + 18*d*s^2 + a*b^2*r*s + 15*a*d*r*s + 9*a^2*d*r^2 + b^3*r^2 - 9*b*d*r^2 + 15*a*b*d*t + 6*b^2*d*s - 24*d^2*s - 18*a*d^2*r + 10*d^3) = 0

These equation clues lead to the complete quartic key for the Adamchik Transformation as a special case of the Tschirnhaus Transformation.

Synthesis of the Bring Jerrard form

The last two mentioned formulas of the old equation system lead to the coefficient of the linear term and to the coefficient of the absolute term of the aimed Bring Jerrard form:

$y^{5}+my+n=0$

$-m=$ -2*r^2*t^2 + 4*r*s^2*t - s^4 - 3*a*r^2*s*t + a*r*s^3 - 5*a*t^3 - a^2*s*t^2 + 2*b*r^3*t - b*r^2*s^2 + 6*b*s*t^2 + 2*a^3*r*t^2 - 4*a*b*r*t^2 + 7*c*r*t^2 + -a^3*s^2*t + 3*a*b*s^2*t + c*r^3*s - 7*c*s^2*t + 3*a^4*r*s*t - 11*a^2*b*r*s*t + 10*a*c*r*s*t + 4*b^2*r*s*t - 16*d*r*s*t - a^4*s^3 + 4*a^2*b*s^3 - 4*a*c*s^3 - 2*b^2*s^3 - 2*d*r^4 + 8*d*s^3 - 2*a^3*b*r^2*t + 6*a*b^2*r^2*t - 2*a*d*r^2*t - 6*b*c*r^2*t + a^3*b*r*s^2 - a^2*c*r*s^2 - 3*a*b^2*r*s^2 + 2*a*d*r*s^2 + 5*b*c*r*s^2 + 5*a^3*c*t^2 - 5*a^2*b^2*t^2 - 10*a^2*d*t^2 + 5*a*b*c*t^2 + 5*b^3*t^2 - 10*b*d*t^2 - 5*c^2*t^2 - a^3*c*r^2*s + 2*a^2*d*r^2*s + 3*a*b*c*r^2*s - 4*b*d*r^2*s - 3*c^2*r^2*s - 6*a^3*d*s*t + 7*a^2*b*c*s*t - a*b^3*s*t + 4*a*b*d*s*t - 13*a*c^2*s*t - 3*b^2*c*s*t + 22*c*d*s*t + 2*a^3*d*r^3 - 6*a*b*d*r^3 + 6*c*d*r^3 + 14*a^2*b*d*r*t - 3*a^2*c^2*r*t - 6*a*b^2*c*r*t - 2*a*c*d*r*t + 2*b^4*r*t - 16*b^2*d*r*t + 9*b*c^2*r*t + 12*d^2*r*t - 8*a^2*b*d*s^2 - 2*a^2*c^2*s^2 + 4*a*b^2*c*s^2 + 16*a*c*d*s^2 - b^4*s^2 + 8*b^2*d*s^2 - 4*b*c^2*s^2 - 18*d^2*s^2 + 10*a^2*c*d*r*s - 2*a*b^2*d*r*s - 3*a*b*c^2*r*s - 15*a*d^2*r*s + b^3*c*r*s - 4*b*c*d*r*s + 3*c^3*r*s - 9*a^2*d^2*r^2 + 6*a*b*c*d*r^2 - 2*b^3*d*r^2 + 9*b*d^2*r^2 - 6*c^2*d*r^2 - 15*a*b*d^2*t + 10*a*c^2*d*t + 10*b^2*c*d*t - 5*b*c^3*t - 15*c*d^2*t - 12*a*c*d^2*s - 6*b^2*d^2*s + 8*b*c^2*d*s - c^4*s + 16*d^3*s + 12*a*d^3*r - 9*b*c*d^2*r + 2*c^3*d*r - 5*d^4

$-n=$ - c s^3 t - a b r s t^2 + 3 c r s t^2 + b s^2 t^2 + a^2 r t^3 - 2 b r t^3 - a s t^3 + t^4 - 2*a*c*r^2*t^2 + a*c*r*s^2*t + b^2*r^2*t^2 + 2*d*r^2*t^2 - 4*d*r*s^2*t + d*s^4 - a^5*t^3 + 5*a^3*b*t^3 - 5*a^2*c*t^3 - 5*a*b^2*t^3 + 3*a*d*r^2*s*t - a*d*r*s^3 + 5*a*d*t^3 - b*c*r^2*s*t + 5*b*c*t^3 + a^4*b*s*t^2 - a^3*c*s*t^2 - 4*a^2*b^2*s*t^2 + a^2*d*s*t^2 + 7*a*b*c*s*t^2 + 2*b^3*s*t^2 - 2*b*d*r^3*t + b*d*r^2*s^2 - 6*b*d*s*t^2 + c^2*r^3*t - 3*c^2*s*t^2 + 2*a^4*c*r*t^2 - a^3*b^2*r*t^2 - 3*a^3*d*r*t^2 - 6*a^2*b*c*r*t^2 + 3*a*b^3*r*t^2 + 4*a*b*d*r*t^2 + 7*a*c^2*r*t^2 - 3*b^2*c*r*t^2 - 7*c*d*r*t^2 - a^4*c*s^2*t + a^3*d*s^2*t + 4*a^2*b*c*s^2*t - 3*a*b*d*s^2*t - 4*a*c^2*s^2*t - 2*b^2*c*s^2*t - c*d*r^3*s + 7*c*d*s^2*t - 3*a^4*d*r*s*t + a^3*b*c*r*s*t + 11*a^2*b*d*r*s*t - a^2*c^2*r*s*t - 3*a*b^2*c*r*s*t - 10*c*a*d*r*s*t - 4*b^2*d*r*s*t + 5*b*c^2*r*s*t + 8*d^2*r*s*t + a^4*d*s^3 - 4*a^2*b*d*s^3 + 4*a*c*d*s^3 + 2*b^2*d*s^3 + d^2*r^4 - 4*d^2*s^3 + 2*a^3*b*d*r^2*t - a^3*c^2*r^2*t - 6*a*b^2*d*r^2*t + 3*a*b*c^2*r^2*t + a*d^2*r^2*t + 6*b*c*d*r^2*t - 3*c^3*r^2*t - a^3*b*d*r*s^2 + a^2*c*d*r*s^2 + 3*a*b^2*d*r*s^2 - a*d^2*r*s^2 - 5*b*c*d*r*s^2 - 5*a^3*c*d*t^2 + 5*a^2*b^2*d*t^2 + 5*a^2*b*c^2*t^2 + 5*a^2*d^2*t^2 - 5*a*b^3*c*t^2 - 5*a*b*c*d*t^2 - 5*a*c^3*t^2 + b^5*t^2 - 5*b^3*d*t^2 + 5*b^2*c^2*t^2 + 5*b*d^2*t^2 + 5*c^2*d*t^2 + a^3*c*d*r^2*s - a^2*d^2*r^2*s - 3*a*b*c*d*r^2*s + 2*b*d^2*r^2*s + 3*c^2*d*r^2*s + 3*a^3*d^2*s*t - 7*a^2*b*c*d*s*t - 2*a^2*c^3*s*t + a*b^3*d*s*t + 4*a*b^2*c^2*s*t - 2*a*b*d^2*s*t + 13*a*c^2*d*s*t - b^4*c*s*t + 3*b^2*c*d*s*t - 4*b*c^3*s*t - 11*c*d^2*s*t - a^3*d^2*r^3 + 3*a*b*d^2*r^3 - 3*c*d^2*r^3 - 7*a^2*b*d^2*r*t + 3*a^2*c^2*d*r*t + 6*a*b^2*c*d*r*t - 3*a*b*c^3*r*t + a*c*d^2*r*t - 2*b^4*d*r*t + b^3*c^2*r*t + 8*b^2*d^2*r*t - 9*b*c^2*d*r*t + 3*c^4*r*t - 4*d^3*r*t + 4*a^2*b*d^2*s^2 + 2*a^2*c^2*d*s^2 - 4*a*b^2*c*d*s^2 - 8*a*c*d^2*s^2 + b^4*d*s^2 - 4*b^2*d^2*s^2 + 4*b*c^2*d*s^2 + 6*d^3*s^2 - 5*a^2*c*d^2*r*s + a*b^2*d^2*r*s + 3*a*b*c^2*d*r*s + 5*a*d^3*r*s - b^3*c*d*r*s + 2*b*c*d^2*r*s - 3*c^3*d*r*s + 3*a^2*d^3*r^2 - 3*a*b*c*d^2*r^2 + b^3*d^2*r^2 - 3*b*d^3*r^2 + 3*c^2*d^2*r^2 + 5*a*b*d^3*t - 5*a*c^2*d^2*t - 5*b^2*c*d^2*t + 5*b*c^3*d*t - c^5*t + 5*c*d^3*t + 4*a*c*d^3*s + 2*b^2*d^3*s - 4*b*c^2*d^2*s + c^4*d*s - 4*d^4*s - 3*a*d^4*r + 3*b*c*d^3*r - c^3*d^2*r + d^5

And so the complete Adamchik Transformation is created.

Calculation example

Determination of the Quartic key

Given is the Principal Quintic equation that can not be solved by elementary roots:

$x^{5}-5x^{2}+5x-5=0$

In this example following pattern is valid:

$r=-5,s=5,t=-5$

The system of three equations and three unknowns a, b and d is set up:

$\mathrm {I^{*}} )\,\,\,20-15a-5d=0$

$\mathrm {II^{*}} )\,\,\,-25+20a-15b=0$

$\mathrm {III^{*}} )\,\,\,25-75ab+30b^{2}+15ad+5d=0$

This equation system has following solution:

$a=-1,b=-3,d=7$

$a=7/5,b=1/5,d=-1/5$

Only the upper solution of that equation system leads to a cubic equation for c that has one real and two imaginary solution. In the other case there appear three real solutions. Therefore exactly the upper solution now will be entered into the mentioned cubic equation:

$-5c^{3}-110c^{2}-1100c-3080=0$

This is idential to that:

$c^{3}+22c^{2}+220c+616=0$

And in this way the real solution for c appears:

$c={\frac {8}{3}}{\sqrt {11}}\sinh {\bigl [}{\frac {1}{3}}\operatorname {arsinh} {\bigl (}{\frac {4}{11}}{\sqrt {11}}{\bigr )}{\bigr ]}-{\frac {22}{3}}$

$c\approx -4.2684949995680718959592849407954973796$

Determination of the Bring Jerrard form coefficients

This solution now is entered into the expressions for the coefficients of the Bring Jerrard form m and n to accurately find out these two coefficients:

The value of m is this:

$m={\frac {14080}{3}}{\bigl \{}11-2{\sqrt {22}}\cosh {\bigl [}{\frac {1}{3}}\operatorname {arcosh} {\bigl (}{\frac {7}{22}}{\sqrt {22}}{\bigr )}{\bigr ]}{\bigr \}}$

$m\approx 5346.7659799413342912673562726724783512142$

Corresponding equation:

$m^{3}-154880m^{2}+6542131200m-30704402432000=0$

And the value of n is that:

$n={\frac {11264}{3}}{\bigl \{}11-110{\sqrt {2}}\sinh {\bigl [}{\frac {1}{3}}\operatorname {arsinh} {\bigl (}{\frac {1}{22}}{\sqrt {2}}{\bigr )}{\bigr ]}{\bigr \}}$

$n\approx 28793.4255440915279846424972250875$

Corresponding equation:

$n^{3}-123904n^{2}+260987420672n-7435869363372032=0$

So the complete Adamchik Transformation for this calculation example is made:

$x^{5}-5x^{2}+5x-5=0$

$y=x^{4}-x^{3}-3x^{2}+{\bigl \{}{\frac {8}{3}}{\sqrt {11}}\sinh {\bigl [}{\frac {1}{3}}\operatorname {arsinh} {\bigl (}{\frac {4}{11}}{\sqrt {11}}{\bigr )}{\bigr ]}-{\frac {22}{3}}{\bigr \}}x+7$

$y^{5}+{\frac {14080}{3}}{\bigl \{}11-2{\sqrt {22}}\cosh {\bigl [}{\frac {1}{3}}\operatorname {arcosh} {\bigl (}{\frac {7}{22}}{\sqrt {22}}{\bigr )}{\bigr ]}{\bigr \}}y+{\frac {11264}{3}}{\bigl \{}11-110{\sqrt {2}}\sinh {\bigl [}{\frac {1}{3}}\operatorname {arsinh} {\bigl (}{\frac {1}{22}}{\sqrt {2}}{\bigr )}{\bigr ]}{\bigr \}}=0$

$y\approx -4.87187987090341241739191116705958390845844658170795795268900739402026742$

$x\approx 1.56670895425072582758152133323240667646412076995364965189840377191745$

This described Adamchik Transformation gives the exact solution of the given Principal Quintic equation.

Shortcut to the Brioschi form

Description of the shortcut

For finding out the corresponding Brioschi form to the given Principal quintic the same equation system delivers a shortcut directly to the coefficients of the broken rational key of the Brioschi form that is described in the Titus-Piezas-III-essay. This is the equation system again. The coefficients a, b and d give the complete transformation of the Principal form into the Brioschi form. This is the given principal form:

$x^{5}+rx^{2}+sx+t=0$

And this is the given equation system:

$\mathrm {I^{*}} )\,\,\,4s+3ar-5d=0$

$\mathrm {II^{*}} )\,\,\,5t+4as+3br=0$

$\mathrm {III^{*}} )\,\,\,3rt-2s^{2}+15abt+6b^{2}s-3adr+ds=0$

Now the coefficients of the broken rational key of the Brioschi form will be created by using the coefficent a in this way:

$u={\frac {(27r^{4}-160s^{3}+300rst)a+30r^{3}s-125s^{2}t}{3r^{4}-15s^{3}+75rst}}$

$v=1728-{\frac {5(15t-ru^{2}+3su)^{3}}{r^{2}(s^{2}u-5rtu+5st)}}$

$w={\frac {5(15t-ru^{2}+3su)^{3}-(s^{2}u-5rtu+5st)(40ru^{3}+360su^{2}+1800tu)}{5(s^{2}u-5rtu+5st)(ru^{2}+su+5t)}}$

This is the broken rational key for the Brioschi form:

$x={\frac {wz+u}{vz^{2}-3}}$

The resulting Brioschi equation looks this way:

$v^{2}z^{5}-10vz^{3}+45z-1=0$

Mentioned example

By taking the already mentioned calculation example following Brioschi form appears:

$x^{5}-5x^{2}+5x-5=0$

The value for a was already computed in the mentioned calculation example:

$a=-1$

This solution brings those values:

$u=-4\,\,{\text{and}}\,\,v=1849\,\,{\text{and}}\,\,w=129$

This is the broken rational key:

$x={\frac {129\,z-4}{1849\,z^{2}-3}}$

It shows the Tschirnhaus Transformation in this way:

${\biggl [}{\frac {129\,z-4}{1849\,z^{2}-3}}{\biggr ]}^{5}-5{\biggl [}{\frac {129\,z-4}{1849\,z^{2}-3}}{\biggr ]}^{2}+5{\biggl [}{\frac {129\,z-4}{1849\,z^{2}-3}}{\biggr ]}-5=0$

And this is the resulting Brioschi equation:

$3418801z^{5}-18490z^{3}+45z-1=0$

Other sources

• Victor Adamchik, David Jeffrey: Archived at sigsam.org , ACM Sigsam Bulletin, Band 37, 2003

• F. Brioschi: Sulla risoluzione delle equazioni del quinto grado: Hermite — Sur la résolution de l'Équation du cinquiéme degré Comptes rendus —. N. 11. Mars. 1858. 1. Dezember 1858, doi:10.1007/bf03197334

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