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== Quartic case ==

=== Tschirnhaus transformation with four clues ===

This is the given quartic equation:

:<math> x^4 - ax^3 + bx^2 - cx + d = 0 </math>

Now this quadratic equation system shall be solved:

:{| class = "wikitable"
|<math> \mathrm{First\, clue} </math>
|<math> at + 4u + a^2 - 2b = 0 </math>
|-
|<math> \mathrm{Second\, clue} </math>
|<math> bt^2 - 6u^2 + abt - 3ct - 2ac + b^2 + 2d = 0 </math>
|-
|<math> \mathrm{Third\, clue} </math>
|<math> v = c t^3 + a c t^2 - 4 d t^2 - 3 a d t + b c t + 4 u^3 - 2 b d + c^2 </math>
|-
|<math> \mathrm{Fourth\, clue} </math>
|<math> w = d t^4 - u^4 + u v + a d t^3 + b d t^2 + c d t + d^2 </math>
|}

And so accurately that ] appears:

:<math> (x^2 + tx + u)^4 - v (x^2 + tx + u) + w = 0 </math>

=== Quartic calculation examples ===

The Tschirnhaus transformation of the equation for the Tetranacci constant contains only rational coefficients:

:<math> x^4 - x^3 - x^2 - x - 1 = 0 </math>

:<math> y = x^2 - 3x </math>

:<math> y^4 - 11y - 41 = 0 </math>

In this way following expression can be made about the Tetranacci constant:

:<math> x^2 - 3x = (\tfrac{41}{3})^{1/4}\sqrt{\sinh\bigl} - </math>

:<math> - (\tfrac{41}{3})^{1/4}\bigl\{\tfrac{11}{4}(\tfrac{3}{41})^{3/4}\sqrt{\operatorname{csch}\bigl} - \sinh\bigl\bigr\}^{1/2} </math>

That calculation example however does contain the element of the square root in the Tschirnhaus transformation:

:<math> x^4 + x^3 + x^2 - x - 1 = 0 </math>

:<math> y = x^2 + \tfrac{1}{5}(19+4\sqrt{21})x + \tfrac{1}{5}(6+\sqrt{21}) </math>

:<math> y^4 - \tfrac{1}{125}(38267 + 8272\sqrt{21})y - \tfrac{1}{625}(101277\sqrt{21} + 463072) = 0 </math>

=== Special form of the quartic ===

In the following we solve a special equation pattern that is easily solvable by using elliptic functions:

:<math> x^4 - 6x^2 - 8\sqrt{S^2+1}\,x - 3 = 0 </math>

:<math> Q = q\bigl\{ \tanh\bigl \bigr\} = q\bigl </math>
:<math> x = \frac{3\,\vartheta_{01}(Q^3)^2}{\vartheta_{01}(Q)^2} </math>

These are important additional informations about the ] and the mentioned Jacobi theta function:

:<math> q(\varepsilon) = \exp\bigl </math>

:<math> \vartheta_{01}(r) = \sum_{k = -\infty}^{\infty} (-1)^k r^{k^2} = \prod_{n = 1}^{\infty} (1-r^{2n})(1-r^{2n-1})^2 </math>

Computation rule for the mentioned theta quotient:

:<math> \frac{3\,\vartheta_{01}\{q^3\}^2}{\vartheta_{01}\{q\}^2} = \sqrt{2\sqrt{\kappa^4 - \kappa^2 + 1} - \kappa^2 + 2} + \sqrt{\kappa^2 + 1} </math>

Accurately the ] is used for solving that equation.

Now we create a Tschirnhaus transformation on that:

:<math> x^4 - 6x^2 - 8\sqrt{S^2+1}\,x - 3 = 0 </math>

:<math> y = x^2-2(\sqrt{S^2+1}-S)x-3 </math>

:<math> y^4 + 64\,S^2 (4S^2+1-4S\sqrt{S^2+1})y - 384\,S^3 (\sqrt{S^2+1}-S) = 0 </math>

=== Elliptic solving of principal quartics ===

Now this solving pattern shall be used for solving some principal quartic equations:

First calculation example:

:<math> x^4 + x - 1 = 0 </math>

:<math> Q = q\bigl\{ \tanh \bigl \bigr\} </math>

:<math> x = \frac{4}{\sqrt{2\sqrt{849}+18}-6} \biggl </math>

Second calculation example:

:<math> x^4 + 2x - 1 = 0 </math>

:<math> Q = q\bigl\{ \tanh \bigl \bigr\} </math>

:<math> x = \frac{2}{\sqrt{2\sqrt{129}+18}-6} \biggl </math>

Third calculation example:

:<math> x^4 + 5x - 3 = 0 </math>

:<math> Q = q\bigl\{ \tanh \bigl \bigr\} </math>

:<math> x = \frac{4}{\sqrt{2\sqrt{881}+50}-10} \biggl </math>

Clue for the computation of the S value for a given principal quartic:

:<math> x^4 + \psi x - \omega = 0 </math>

If this equation pattern is given, the modulus tangent duplication value S can be determined in this way:

:<math> \psi^4 \bigl^3 = \omega^3 \bigl^4 </math>

The solution of the now mentioned formula always is in pure biquadratic radical relation to psi and omega and therefore it is a useful tool to solve principal quartic equations.

:<math> Q = \exp\bigl\langle - \pi K\bigl\{ \operatorname{sech}\bigl \bigr\} \div K\bigl\{ \tanh\bigl \bigr\} \bigr\rangle </math>

And this can be solved in that way:

:<math> x = \frac{\omega }{\psi } \biggl </math>


== Quintic case == == Quintic case ==

Revision as of 13:53, 30 December 2024

In mathematics and, more specifically, in theory of equations, the principal form of an irreducible polynomial of degree at least three is a polynomial of the same degree n without terms of degrees n−1 and n−2, such that each roots of either polynomial is a rational function of a root of the other polynomial.

The principal form of a polynomial can be found by applying a suitable Tschirnhaus transformation to the given polynomial.

Definition

Let

f ( x ) = x n + a 1 x n − 1 + ⋯ + a n − 1 x + a n {\displaystyle f(x)=x^{n}+a_{1}x^{n-1}+\cdots +a_{n-1}x+a_{n}}

be an irreducible polynomial of degree at least three.

Its principal form is a polynomial

g ( y ) = y n + b 3 y n − 3 + ⋯ + b n − 1 y + b n , {\displaystyle g(y)=y^{n}+b_{3}y^{n-3}+\cdots +b_{n-1}y+b_{n},}

together with a Tschirnhaus transformation of degree two

φ ( x ) = x 2 + α x + β {\displaystyle \varphi (x)=x^{2}+\alpha x+\beta }

such that, if r is a root of f, ϕ ( r ) {\displaystyle \phi (r)} is a root of ⁠ g {\displaystyle g} ⁠.

Expressing that ⁠ g {\displaystyle g} ⁠ does not has terms in ⁠ y n − 1 {\displaystyle y^{n-1}} ⁠ and ⁠ y n − 2 {\displaystyle y^{n-2}} ⁠ leads to a system of two equations in ⁠ α {\displaystyle \alpha } ⁠ and ⁠ β {\displaystyle \beta } ⁠, one of degree one and one of degree two. In general, this system has two solutions, giving two principal forms involving a square root. One passes from one principal form to the secong by changing the sign of the square root.

Quintic case

Synthesis advice for the quadratic Tschirnhaus key

This is the given quintic equation:

x 5 − a x 4 + b x 3 − c x 2 + d x − e = 0 {\displaystyle x^{5}-ax^{4}+bx^{3}-cx^{2}+dx-e=0}

That quadratic equation system leads to the coefficients of the quadratic Tschirnhaus key:

F i r s t c l u e {\displaystyle \mathrm {First\,clue} } a s + 5 t + a 2 − 2 b = 0 {\displaystyle as+5t+a^{2}-2b=0}
S e c o n d c l u e {\displaystyle \mathrm {Second\,clue} } b s 2 − 10 t 2 + a b s − 3 c s − 2 a c + b 2 + 2 d = 0 {\displaystyle bs^{2}-10t^{2}+abs-3cs-2ac+b^{2}+2d=0}

By polynomial division that Tschirnhaus transformation can be made:

( x 2 + s x + t ) 5 − u ( x 2 + s x + t ) 2 + v ( x 2 + s x + t ) − w = 0 {\displaystyle (x^{2}+sx+t)^{5}-u(x^{2}+sx+t)^{2}+v(x^{2}+sx+t)-w=0}

Calculation examples

This is the first example:

x 5 − x 4 − x 2 − 1 = 0 {\displaystyle x^{5}-x^{4}-x^{2}-1=0}
y = x 2 − 1 4 ( 19 − 265 ) x − 1 20 ( 265 − 15 ) {\displaystyle y=x^{2}-{\tfrac {1}{4}}(19-{\sqrt {265}})x-{\tfrac {1}{20}}({\sqrt {265}}-15)}
y 5 + 1 80 ( 24455 − 1501 265 ) y 2 − 1 160 ( 5789 265 − 93879 ) y − 1 4000 ( 5393003 265 − 87785025 ) = 0 {\displaystyle y^{5}+{\tfrac {1}{80}}(24455-1501{\sqrt {265}})y^{2}-{\tfrac {1}{160}}(5789{\sqrt {265}}-93879)y-{\tfrac {1}{4000}}(5393003{\sqrt {265}}-87785025)=0}

And this is the second example:

x 5 + x 4 + x 3 + x 2 − 1 = 0 {\displaystyle x^{5}+x^{4}+x^{3}+x^{2}-1=0}
y = x 2 + 1 3 ( 30 − 3 ) x + 1 15 30 {\displaystyle y=x^{2}+{\tfrac {1}{3}}({\sqrt {30}}-3)x+{\tfrac {1}{15}}{\sqrt {30}}}
y 5 − 1 45 ( 465 − 61 30 ) y 2 + 2 45 ( 1616 − 289 30 ) y − 1 1125 ( 33758 30 − 183825 ) = 0 {\displaystyle y^{5}-{\tfrac {1}{45}}(465-61{\sqrt {30}})y^{2}+{\tfrac {2}{45}}(1616-289{\sqrt {30}})y-{\tfrac {1}{1125}}(33758{\sqrt {30}}-183825)=0}

Solving the principal quintic via Adamchik transformation

The mathematicians Victor Adamchik and David Jeffrey found out how to solve every principal quintic equation. In their essay Polynomial Transformations of Tschirnhaus, Bring and Jerrard they wrote this way down. These two mathematicians solved this principal form by transforming it into the Bring Jerrard form. Their method contains the construction of a quartic Tschirnhaus transformation key. For the construction of that key they executed a disjunction of the linear term coefficient of the key in order to get a system that solves all other terms in a quadratic radical way and to only solve a further cubic equation to get the coefficient of the linear term of the Tschirnhaus key.

In their essay they constructed the quartic Tschirnhaus key in this way:

y 5 − u y 2 + v y − w = 0 {\displaystyle y^{5}-uy^{2}+vy-w=0}

In order to do the transformation Adamchik and Jeffrey constructed equation system that generates the coefficients of the cubic, quadratic and absolute term of the Tschirnhaus key:

4 v α − 3 u β = 5 w {\displaystyle 4v{\color {crimson}\alpha }-3u{\color {green}\beta }=5w}
3 u α + 5 δ = 4 v {\displaystyle 3u{\color {crimson}\alpha }+5{\color {blue}\delta }=4v}
15 w α β − 3 u α δ − 6 v β 2 − v δ = 3 u w − 2 v 2 {\displaystyle 15w{\color {crimson}\alpha }{\color {green}\beta }-3u{\color {crimson}\alpha }{\color {blue}\delta }-6v{\color {green}\beta }^{2}-v{\color {blue}\delta }=3uw-2v^{2}}

And for receiving the coefficient of the linear term this cubic equation shall be solved successively:

u γ 3 + ( 5 w α − 4 v β + 3 u 2 ) γ 2 + ( u v α 2 − 8 v α δ + 5 w β 2 + 6 u β δ − 8 u v β + 3 u 3 + 9 v w ) γ + {\displaystyle u{\color {orange}\gamma }^{3}+(5w{\color {crimson}\alpha }-4v{\color {green}\beta }+3u^{2}){\color {orange}\gamma }^{2}+(uv{\color {crimson}\alpha }^{2}-8v{\color {crimson}\alpha }{\color {blue}\delta }+5w{\color {green}\beta }^{2}+6u{\color {green}\beta }{\color {blue}\delta }-8uv{\color {green}\beta }+3u^{3}+9vw){\color {orange}\gamma }+}
+ u 3 α 3 + v w α 3 + u 2 β 3 − 10 u w α 2 β + 4 v 2 α 2 β + 9 u 2 α 2 δ − u v α β 2 + 15 u α δ 2 + 10 δ 3 − {\displaystyle +u^{3}{\color {crimson}\alpha }^{3}+vw{\color {crimson}\alpha }^{3}+u^{2}{\color {green}\beta }^{3}-10uw{\color {crimson}\alpha }^{2}{\color {green}\beta }+4v^{2}{\color {crimson}\alpha }^{2}{\color {green}\beta }+9u^{2}{\color {crimson}\alpha }^{2}{\color {blue}\delta }-uv{\color {crimson}\alpha }{\color {green}\beta }^{2}+15u{\color {crimson}\alpha }{\color {blue}\delta }^{2}+10{\color {blue}\delta }^{3}-}
− u 2 v α 2 − 3 u 3 α β + 6 v w α β − 12 u v α δ + 5 u w β 2 − 4 v 2 β 2 + u v 2 α − u 2 w α + 2 u 2 v β + u 4 − 4 v 3 + 8 u v w = 0 {\displaystyle -u^{2}v{\color {crimson}\alpha }^{2}-3u^{3}{\color {crimson}\alpha }{\color {green}\beta }+6vw{\color {crimson}\alpha }{\color {green}\beta }-12uv{\color {crimson}\alpha }{\color {blue}\delta }+5uw{\color {green}\beta }^{2}-4v^{2}{\color {green}\beta }^{2}+uv^{2}{\color {crimson}\alpha }-u^{2}w{\color {crimson}\alpha }+2u^{2}v{\color {green}\beta }+u^{4}-4v^{3}+8uvw=0}

The solution of that system then has to be entered in that mold here:

z = y 4 + α y 3 + β y 2 + γ y + δ {\displaystyle z=y^{4}+{\color {crimson}\alpha }y^{3}+{\color {green}\beta }y^{2}+{\color {orange}\gamma }y+{\color {blue}\delta }}
z 5 + λ z − μ = 0 {\displaystyle z^{5}+\lambda z-\mu =0}

The coefficients Lambda and My can be found out by doing a polynomial division of z^5 divided by the initial principal polynome and reading the resulting remainder rest. So a Bring Jerrard equation appears that contains only the quintic, the linear and the absolute term. That equation can be solved by an elliptic Jacobi theta quotient that contains the fifth powers and the fifth roots of the corresponding elliptic nome in the theta function terms.

Examples of solving the principal form

Along with the Abel Ruffini theorem the following equation is an example that can not be solved in an elementary way, but can be reduced to the Bring Jerrard form by only using cubic radical elements. This shall be demonstrated here:

y 5 − 5 y 2 + 5 y − 5 = 0 {\displaystyle y^{5}-5y^{2}+5y-5=0}

For this given equation the quartic Tschirnhaus key now shall be synthesized. For that example the values u = v = w = 5 are the combination. Therefore, this is the mentioned equation system for that example:

20 α − 15 β = 25 {\displaystyle 20{\color {crimson}\alpha }-15{\color {green}\beta }=25}
15 α + 5 δ = 20 {\displaystyle 15{\color {crimson}\alpha }+5{\color {blue}\delta }=20}
75 α β − 15 α δ − 30 β 2 − 5 δ = 25 {\displaystyle 75{\color {crimson}\alpha }{\color {green}\beta }-15{\color {crimson}\alpha }{\color {blue}\delta }-30{\color {green}\beta }^{2}-5{\color {blue}\delta }=25}

So these are the coefficients of the cubic, quadratic and absolute term of the Tschirnhaus transformation key:

α = − 1 , β = − 3 , δ = 7 {\displaystyle {\color {crimson}\alpha }=-1,\,{\color {green}\beta }=-3,\,{\color {blue}\delta }=7}

That cubic equation leads to the coefficient of the linear term of the key:

5 γ 3 + 110 γ 2 + 1100 γ + 3080 = 0 {\displaystyle 5{\color {orange}\gamma }^{3}+110{\color {orange}\gamma }^{2}+1100{\color {orange}\gamma }+3080=0}
γ = 8 3 11 sinh ⁡ [ 1 3 arsinh ⁡ ( 4 11 11 ) ] − 22 3 {\displaystyle {\color {orange}\gamma }={\frac {8}{3}}{\sqrt {11}}\sinh {\bigl }-{\frac {22}{3}}}

By doing a polynomial division on the fifth power of the quartic Tschirnhaus transformation key and analyzing the remainder rest the coefficients of the mold can be determined. This is the result:

y 5 − 5 y 2 + 5 y − 5 = 0 {\displaystyle y^{5}-5y^{2}+5y-5=0}
z = y 4 − y 3 − 3 y 2 + { 8 3 11 sinh ⁡ [ 1 3 arsinh ⁡ ( 4 11 11 ) ] − 22 3 } y + 7 {\displaystyle z=y^{4}-y^{3}-3y^{2}+{\bigl \{}{\frac {8}{3}}{\sqrt {11}}\sinh {\bigl }-{\frac {22}{3}}{\bigr \}}y+7}
z 5 + 14080 3 { 11 − 2 22 cosh ⁡ [ 1 3 arcosh ⁡ ( 7 22 22 ) ] } z + 11264 3 { 11 − 110 2 sinh ⁡ [ 1 3 arsinh ⁡ ( 1 22 2 ) ] } = 0 {\displaystyle z^{5}+{\frac {14080}{3}}{\bigl \{}11-2{\sqrt {22}}\cosh {\bigl }{\bigr \}}z+{\frac {11264}{3}}{\bigl \{}11-110{\sqrt {2}}\sinh {\bigl }{\bigr \}}=0}

These are the approximations of the solution:

z ≈ − 4.87187987090341241739191116705958390845844658170795795268900739402026742 {\displaystyle z\approx -4.87187987090341241739191116705958390845844658170795795268900739402026742}
y ≈ 1.56670895425072582758152133323240667646412076995364965189840377191745 {\displaystyle y\approx 1.56670895425072582758152133323240667646412076995364965189840377191745}

Literature

References

  1. Weisstein, Eric W. "Principal Quintic Form". mathworld.wolfram.com.
  2. "The solution to the principal quintic via the Brioschi and Rogers-Ramanujan cfrac $R(q)$". Mathematics Stack Exchange.
  3. Jerrard, George Birch (1859). An essay on the resolution of equations. London, UK: Taylor & Francis.
  4. Adamchik, Victor (2003). "Polynomial Transformations of Tschirnhaus, Bring, and Jerrard" (PDF). ACM SIGSAM Bulletin. 37 (3): 91. CiteSeerX 10.1.1.10.9463. doi:10.1145/990353.990371. S2CID 53229404. Archived from the original (PDF) on 2009-02-26.
  5. Victor S. Adamchik and David J. Jeffrey. "Polynomial Transformations of Tschirnhaus, Bring and Jerrard" (PDF). ACM SIGSAM Bulletin, Vol 37, No. 3, September 2003. Retrieved 28 December 2024.
  6. "A new way to solve the Bring quintic?". Mathematics Stack Exchange.
  7. Titus Piezas III. ""A New Way To Derive The Bring-Jerrard Quintic In Radicals"". oocities.org. Retrieved 28 December 2024.
  8. Klein, Felix (December 28, 1888). "Lectures on the ikosahedron and the solution of equations of the fifth degree". London : Trübner & Co. – via Internet Archive.
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