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.

Also see

 * Definition:Depressed Polynomial
 * Tschirnhaus Transformation yields Depressed Polynomial