thumb|[[Ehrenfried Walther von Tschirnhaus]]
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.
Simply, it is a method for transforming a polynomial equation of degree <math>n\ge2</math> with some nonzero intermediate coefficients, <math>a_1, ..., a_{n-1}</math>, such that some or all of the transformed intermediate coefficients, <math>a'_1, ..., a'_{n-1}</math>, are exactly zero.
For example, finding a substitution<math display="block">y(x)=k_1x^2 + k_2x+k_3</math>for a cubic equation of degree <math>n=3</math>,<math display="block">f(x) = x^3+a_2x^2+a_1x+a_0</math>such that substituting <math>x=x(y)</math> yields a new equation<math display="block">f'(y)=y^3+a'_2y^2+a'_1y+a'_0</math>such that <math>a'_1=0</math>, <math>a'_2=0</math>, 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 <math>n^{th}</math> degree reducible monic polynomial equation <math>f(x)=0</math> of the form <math>f(x) = g(x) / h(x)</math>, where <math>g(x)</math> and <math>h(x)</math> are polynomials and <math>h(x)</math> does not vanish at <math>f(x) = 0</math>,<math display="block">f(x) = x^n+a_1x^{n-1}+a_2x^{n-2}+...+a_{n-1}x+a_n=0</math>the Tschirnhaus transformation is the function:<math display="block">y=k_1x^{n-1} + k_2x^{n-2}+...+k_{n-1}x+k_n</math>Such that the new equation in <math>y</math>, <math>f'(y)</math>, has certain special properties, most commonly such that some coefficients, <math>a'_1,...,a'_{n-1}</math>, are identically zero.
Example: Tschirnhaus' method for cubic equations
In Tschirnhaus' 1683 paper, In his paper, Tschirnhaus referenced a method by René Descartes to reduce a quadratic polynomial <math>(n=2)</math> such that the <math>x</math> term has zero coefficient.
In 1786, this work was expanded by Erland Samuel Bring who showed that any generic quintic polynomial could be similarly reduced.
In 1834, George Jerrard further expanded Tschirnhaus' work by showing a Tschirnhaus transformation may be used to eliminate the <math>x^{n-1}</math>, <math>x^{n-2}</math>, and <math>x^{n-3}</math> for a general polynomial of degree <math>n>3</math>.