Tschirnhaus transformation

A Tschirnhaus transformation (also Tschirnhausen transformation) is a variable transformation that makes it possible to simplify algebraic equations of a higher degree.

They were introduced by Ehrenfried Walther von Tschirnhaus in 1683 (published in the Acta Eruditorum ).

description

The -th degree equation ${\displaystyle n}$

${\displaystyle f(x)=x^{n}+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+a_{n-3}x^{n-3}+\dotsb +a_{1}x+a_{0}=0}$,

is made by a variable transformation of the form

${\displaystyle y=c_{m-1}x^{m-1}+c_{m-2}x^{m-2}+\dotsb +c_{1}x+c_{0}}$

on the shape

${\displaystyle g(y)=y^{n}+b_{n-1}y^{n-1}+b_{n-2}y^{n-2}+b_{n-3}y^{n-3}+\dotsb +b_{1}y+b_{0}=0}$

brought.

The aim is to choose the coefficients so skillfully that some of the coefficients disappear, that is, equal to 0. ${\displaystyle c_{i}}$ ${\displaystyle b_{i}}$

Calculating the transformed equation

The determination of the coefficients of the transformed equation is generally possible because the coefficients are symmetrical functions in the solutions of the equation . Therefore, the coefficients can be expressed polynomially by the elementary symmetric functions in these solutions. ${\displaystyle b_{i}}$ ${\displaystyle b_{i}}$ ${\displaystyle f(x)=0}$ ${\displaystyle b_{i}}$

Applications

Linear Tschirnhaus transformation

Even before Tschirnhaus it was known that the general cubic equation can be reduced to a normal form without a quadratic term by a linear transformation of the variables (see cubic equation ).

Analogously, for every equation -th degree, the coefficient of the second highest power, i.e., can be made to disappear by a linear transformation . ${\displaystyle n}$ ${\displaystyle b_{n-1}}$ ${\displaystyle y=x+a_{n-1}/n}$

Square Tschirnhaus transformation

Tschirnhaus showed that a cubic equation can be transformed into a form using a quadratic transformation . ${\displaystyle y^{3}-b=0}$

Tschirnhaus therefore said that he had found a general solution method for all algebraic equations, but was taught otherwise by Gottfried Wilhelm Leibniz . Such transformations do not help in solving algebraic equations higher than fourth degree. The reason is that although you can make the coefficients for disappear by choosing which , this leads to a complicated system of equations of different degrees for determining suitable transformation coefficients . In the end, this results in an equation of degree (as Bezout showed). This can still be solved for, but becomes very unwieldy for higher ones . ${\displaystyle c_{i}}$ ${\displaystyle b_{i}}$ ${\displaystyle i=n-1,\dotsc ,1}$ ${\displaystyle c_{i}}$ ${\displaystyle (n-1)!}$ ${\displaystyle n=4}$ ${\displaystyle n}$

In general, in every algebraic equation -th degree, the coefficients can be made to power and to disappear (unless ): First, the coefficient to power is made to disappear by a linear transformation and then the coefficients to the powers and by a quadratic transformation . To determine suitable transformation coefficients, at most a square root must be calculated based on the equation coefficients. ${\displaystyle n}$ ${\displaystyle y^{n-1}}$ ${\displaystyle y^{n-2}}$ ${\displaystyle n>2}$ ${\displaystyle y^{n-1}}$ ${\displaystyle y^{n-1}}$ ${\displaystyle y^{n-2}}$

Higher Tschirnhaus transformations

The coefficient to the power (where is) can also be made to disappear, as Erland Samuel Bring (Lund, 1786) first showed specifically for the quintic . It can be applied to the form with a fourth-degree Tschirnhaus transformation ${\displaystyle y^{n-3}}$ ${\displaystyle n>3}$

${\displaystyle y^{5}+ay+b=0}$

bring (Bring-Jerrard form), and George Jerrard showed in 1834 in general for polynomial equations higher than third degree that one can bring the coefficients to the powers and to vanish by a variable transformation of the fourth degree (at most cube roots and square roots appear in the coefficients on). ${\displaystyle y^{n-1},y^{n-2}}$ ${\displaystyle y^{n-3}}$

When determining the coefficients of the transformation, use is made of the fact that the coefficients or the two equations are given as elementary symmetrical functions by the respective roots of the equations. The elementary symmetric functions are in turn related to the power sums of the roots via the Newton identities . ${\displaystyle c_{i}}$ ${\displaystyle a_{i}}$ ${\displaystyle b_{i}}$ ${\displaystyle f,g}$

Modifications

Modifications of the method have been investigated by Charles Hermite and Arthur Cayley , and Abhyankar emphasized the usefulness of Tschirnhaus's approach in the theory of the resolution of singularities. and uses a generalization of the transformation in the proof of Abhyankar and Moh's theorem.

Web links

• Encyclopedia of Mathematics, Springer
• Tschirnhausen Transformation, Mathworld
• Victor Adamchik, David Jeffrey Polynomial Transformations of Tschirnhaus, Bring and Jerrard , ACM Sigsam Bulletin, Volume 37, 2003, p. 90, pdf

Individual evidence

1. ↑ general, it may also be recognized with polynomials , where and have no common zeros ${\displaystyle y={\tfrac {g(x)}{h(x)}}}$ ${\displaystyle g}$ ${\displaystyle h}$ ${\displaystyle f}$ ${\displaystyle h}$
2. ↑ See Heinrich Weber Algebra , Volume 1, Paragraph 59
3. ↑ Discussion of the method z. B. in Jean-Pierre Tignol, Galois theory of algebraic equations , World Scientific 2001, p. 67
4. ↑ Described e.g. B. in Leonard Dickson , Modern algebraic theories , New York: Sanborn, 1926 (Reprint Dover Phoenix 2004), § 117, or Jörg Bewersdorff , Algebra für Einsteiger , Vieweg 2007, p. 97 f.
5. ^ Dickson, loc. cit., Paragraph 119, p. 212 f.
6. ↑ Hermite Sur quelque théorèmes d'algébre et la résolution de l'équation du quatrième degrée , Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences, Volume 46, 1859, pp. 961-967 ( online ); see Weber, Textbook of Algebra , Volume 1, p. 240, Paragraph 74
7. ↑ On Tschirnhausen's transformation , Philosophical Transactions of the Royal Society London, Volume 152, 1862, pp. 561-578 ( JSTOR 108842 )
8. ↑ See Weber's textbook on algebra
9. ↑ Abhyankar Historical Ramblings in Algebraic Geometry and related algebra , American Mathematical Monthly, June / July 1976, pdf ( Memento of the original from December 31, 2010 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice.