Definition:Tschirnhaus Transformation

Definition

Let $\map f x$ be a polynomial over a field $k$:

$\map f x = a_n x^n + a_{n - 1} x^{n - 1} + a_{n - 2} x^{n - 2} + \cdots + a_1 x + a_0$

Then the Tschirnhaus transformation is the linear substitution $x = y - \dfrac {a_{n - 1} } {n a_n}$.

The Tschirnhaus transformation produces a resulting polynomial $\map {f'} y$ which is depressed, as shown on Tschirnhaus Transformation yields Depressed Polynomial.

This technique is used in the derivation of Cardano's Formula for the roots of the general cubic.

This article is complete as far as it goes, but it could do with expansion. In particular: Brief research suggests that there are more types of Tschirnhaus transformations than just this one. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding this information. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Expand}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Also known as

A Tschirnhaus transformation is also called a Tschirnhaus substitution.

Many sources refer to it as a Tschirnhausen transformation.

Also see

• Definition:Depressed Polynomial
• Tschirnhaus Transformation yields Depressed Polynomial

• Results about Tschirnhaus transformations can be found here.

Source of Name

This entry was named for Ehrenfried Walther von Tschirnhaus.

Sources

• Weisstein, Eric W. "Tschirnhausen Transformation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/TschirnhausenTransformation.html