Category:Definitions/Tschirnhaus Transformations

This category contains definitions related to Tschirnhaus Transformations.
Related results can be found in Category:Tschirnhaus Transformations.

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.