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Revision as of 14:15, 30 December 2024 editD.Lazard (talk | contribs)Extended confirmed users33,906 edits →Solving the principal quintic via Adamchik transformation: Adamchik is not the author of these results that date of the 18th century and are better described in Bring–Jerrard normal formTag: Reverted← Previous edit Revision as of 14:19, 30 December 2024 edit undoD.Lazard (talk | contribs)Extended confirmed users33,906 edits →Quintic case: Huge formulas that are neither explained no sourced. Rm per WP:ORTag: RevertedNext edit →
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== Quintic case ==

=== Synthesis advice for the quadratic Tschirnhaus key ===

This is the given quintic equation:

:<math> x^5 - ax^4 + bx^3 - cx^2 + dx - e = 0 </math>

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

:{| class = "wikitable"
|<math> \mathrm{First\, clue} </math>
|<math> as + 5t + a^2 - 2b = 0 </math>
|-
|<math> \mathrm{Second\, clue} </math>
|<math> bs^2 - 10t^2 + abs - 3cs - 2ac + b^2 + 2d = 0 </math>
|}
By polynomial division that ] can be made:

:<math> (x^2 + sx + t)^5 - u (x^2 + sx + t)^2 + v (x^2 + sx + t) - w = 0 </math>

=== Calculation examples ===

This is the first example:

:<math> x^5 - x^4 - x^2 - 1 = 0 </math>

:<math> y = x^2-\tfrac{1}{4}(19-\sqrt{265})x-\tfrac{1}{20}(\sqrt{265}-15) </math>

:<math> 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 </math>

And this is the second example:

:<math> x^5 + x^4 + x^3 + x^2 - 1 = 0 </math>

:<math> y = x^2 + \tfrac{1}{3}(\sqrt{30}-3)x + \tfrac{1}{15}\sqrt{30} </math>

:<math> 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 </math>

=== Examples of solving the principal form ===

Along with the ] the following equation is an example that can not be solved in an elementary way, but can be reduced<ref>{{Cite web|url=https://archive.org/details/cu31924059413439/page/n181/mode/2up|title=Lectures on the ikosahedron and the solution of equations of the fifth degree|first=Felix|last=Klein|date=December 28, 1888|publisher=London : Trübner & Co.|via=Internet Archive}}</ref> to the ] by only using cubic radical elements. This shall be demonstrated here:

:<math> y^5 - 5y^2 + 5y - 5 =0 </math>

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:

:<math> 20 {\color{crimson}\alpha} - 15 {\color{green}\beta} = 25 </math>

:<math> 15 {\color{crimson}\alpha} + 5 {\color{blue}\delta} = 20 </math>

:<math> 75 {\color{crimson}\alpha} {\color{green}\beta} - 15 {\color{crimson}\alpha} {\color{blue}\delta} - 30 {\color{green}\beta}^2 - 5 {\color{blue}\delta} = 25 </math>

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

:<math> {\color{crimson}\alpha} = -1, \,{\color{green}\beta} = -3, \,{\color{blue}\delta} = 7 </math>

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

:<math> 5{\color{orange}\gamma}^3 + 110{\color{orange}\gamma}^2 + 1100{\color{orange}\gamma} + 3080 = 0 </math>

:<math> {\color{orange}\gamma} = \frac{8}{3}\sqrt{11}\sinh\bigl - \frac{22}{3} </math>

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:

:<math> y^5 - 5y^2 + 5y - 5 =0 </math>

:<math> z = y^4 - y^3 - 3y^2 + \bigl\{\frac{8}{3}\sqrt{11}\sinh\bigl - \frac{22}{3}\bigr\}y + 7 </math>

:<math> 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 </math>

These are the approximations of the solution:

:<math> z \approx -4.87187987090341241739191116705958390845844658170795795268900739402026742 </math>

:<math> y \approx 1.56670895425072582758152133323240667646412076995364965189840377191745 </math>


== Literature == == Literature ==

Revision as of 14:19, 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.

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.
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