Tschirnhaus-Transformation

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

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

Description

The equation of degree

,

Is a variable transformation of the form

To form

Brought.

The goal is to select the coefficients so cleverly that some of the coefficients vanish, that is equal to 0.

Calculation of the transformed equation

Determining the coefficients of the transformed equation is generally possible because the coefficients are symmetrical functions in the solutions of the equation. Therefore, the polynomial coefficients can be expressed by the elementary symmetric functions in these solutions.

Applications

Linear Tschirnhaus transformation

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

Similarly, for each equation of degree, the coefficient of the second highest power, thus, be made ​​to disappear by a linear transformation.

Square Tschirnhaus transformation

Tschirnhaus showed that a cubic equation may be placed on a mold by means of a quadratic transformation.

Tschirnhausen said, therefore, that he had thus found a general solution method for all algebraic equations, but was already taught by Gottfried Wilhelm Leibniz better. Such transformations do not help in solving algebraic equations higher than fourth degree. The reason is that although it can bring the coefficients for the disappearance of choice, but this leads to a complicated system of equations of different degrees to determine an appropriate transform coefficients. This creates at the end of an equation of degree (such as Bezout showed ). This is still solvable for n = 4, but is very cumbersome for higher n

In general, it can be as in any algebraic equation of degree the coefficients to the powers and the disappearance bring (whether n> 2): First, bringing the coefficients for power by a linear transformation to disappear, and then the coefficients and the magnitude of the by a quadratic transformation. To determine a suitable transform coefficients has to be calculated from the equation coefficients most one square root.

Higher Tschirnhaus transformations

The coefficient to the power (where n > 3, it ) can also be made ​​to disappear, as first showed Erland Samuel Bring (Lund, 1786) specifically for the Quintik. It can be a Tschirnhaus transformation of the fourth degree on the shape

Bring ( Bring- Jerrard form), and George Jerrard had 1834 general for polynomial equations higher than the third degree after that you can bring the coefficients to the powers and disappear through a transformation of variables of the fourth degree ( these encounter at most cube roots and square roots in the coefficients on ).

In the determination of the coefficients of the transformation use is made of, and the coefficients of the two equations f, g are added as the elementary symmetric function of the respective roots of the equations. The elementary symmetric functions are again on the Newton identities with the roots of the power sums in connection.

Modifications

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