In mathematics, a Tschirnhaus transformation, also known as Tschirnhausen transformation, is a type of mapping on polynomials developed by Ehrenfried Walther von Tschirnhaus in 1683.^[1]

Simply, it is a method for transforming a polynomial equation of degree ${\displaystyle n\geq 2}$ with some nonzero intermediate coefficients, ${\displaystyle a_{1},...,a_{n-1}}$, such that some or all of the transformed intermediate coefficients, ${\displaystyle a'_{1},...,a'_{n-1}}$, are exactly zero.

For example, finding a substitution

for a cubic equation of degree ${\displaystyle n=3}$,such that substituting ${\displaystyle x=x(y)}$ yields a new equationsuch that ${\displaystyle a'_{1}=0}$, ${\displaystyle a'_{2}=0}$, or both.

More generally, it may be defined conveniently by means of field theory, as the transformation on minimal polynomials implied by a different choice of primitive element. This is the most general transformation of an irreducible polynomial that takes a root to some rational function applied to that root.

Definition

For a generic ${\displaystyle n^{th}}$ degree reducible monic polynomial equation ${\displaystyle f(x)=0}$ of the form ${\displaystyle f(x)=g(x)/h(x)}$, where ${\displaystyle g(x)}$ and ${\displaystyle h(x)}$ are polynomials and ${\displaystyle h(x)}$ does not vanish at ${\displaystyle f(x)=0}$,

the Tschirnhaus transformation is the function:Such that the new equation in ${\displaystyle y}$, ${\displaystyle f'(y)}$, has certain special properties, most commonly such that some coefficients, ${\displaystyle a'_{1},...,a'_{n-1}}$, are identically zero.^[2][3]

Example: Tschirnhaus' method for cubic equations

In Tschirnhaus' 1683 paper,^[1] he solved the equation

using the Tschirnhaus transformation Substituting yields the transformed equation or