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.

Also known as

The Tschirnhaus transformation is also called the Tschirnhaus substitution.

Source of Name

This entry was named for Ehrenfried Walther von Tschirnhaus.