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 see

 * Definition:Depressed Polynomial
 * Tschirnhaus Transformation yields Depressed Polynomial