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Tschirnhaus transformation

## Summary

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

${\displaystyle y(x)=k_{1}x^{2}+k_{2}x+k_{3}}$
for a cubic equation of degree ${\displaystyle n=3}$,
${\displaystyle f(x)=x^{3}+a_{2}x^{2}+a_{1}x+a_{0}}$
such that substituting ${\displaystyle x=x(y)}$ yields a new equation
${\displaystyle f'(y)=y^{3}+a'_{2}y^{2}+a'_{1}y+a'_{0}}$
such 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 ${\displaystyle f(x)=0}$ ,

${\displaystyle f(x)=x^{n}+a_{1}x^{n-1}+a_{2}x^{n-2}+...+a_{n-1}x+a_{n}=0}$

the Tschirnhaus transformation is the function:
${\displaystyle y=k_{1}x^{n-1}+k_{2}x^{n-2}+...+k_{n-1}x+k_{n}}$

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

${\displaystyle f(x)=x^{3}-px^{2}+qx-r=0}$

using the Tschirnhaus transformation
${\displaystyle y(x;a)=x-a\longleftrightarrow x(y;a)=x=y+a.}$

Substituting yields the transformed equation
${\displaystyle f'(y;a)=y^{3}+(3a-p)y^{2}+(3a^{2}-2pa+q)y+(a^{3}-pa^{2}+qa-r)=0}$

or
${\displaystyle {\begin{cases}a'_{1}=3a-p\\a'_{2}=3a^{2}-2pa+q\\a'_{3}=a^{3}-pa^{2}+qa-r\end{cases}}.}$

Setting ${\displaystyle a'_{1}=0}$  yields,
${\displaystyle 3a-p=0\rightarrow a={\frac {p}{3}}}$

and finally the Tschirnhaus transformation
${\displaystyle y=x+{\frac {p}{3}},}$

Which may be substituted into ${\displaystyle f'(y;a)}$  to yield an equation of the form:
${\displaystyle f'(y)=y^{3}-q'y-r'.}$

Tschirnhaus went on to describe how a Tschirnhaus transformation of the form:
${\displaystyle x^{2}(y;a,b)=x^{2}=bx+y+a}$

may be used to eliminate two coefficients in a similar way.

## Generalization

In detail, let ${\displaystyle K}$  be a field, and ${\displaystyle P(t)}$  a polynomial over ${\displaystyle K}$ . If ${\displaystyle P}$  is irreducible, then the quotient ring of the polynomial ring ${\displaystyle K[t]}$  by the principal ideal generated by ${\displaystyle P}$ ,

${\displaystyle K[t]/(P(t))=L}$ ,

is a field extension of ${\displaystyle K}$ . We have

${\displaystyle L=K(\alpha )}$

where ${\displaystyle \alpha }$  is ${\displaystyle t}$  modulo ${\displaystyle (P)}$ . That is, any element of ${\displaystyle L}$  is a polynomial in ${\displaystyle \alpha }$ , which is thus a primitive element of ${\displaystyle L}$ . There will be other choices ${\displaystyle \beta }$  of primitive element in ${\displaystyle L}$ : for any such choice of ${\displaystyle \beta }$  we will have by definition:

${\displaystyle \beta =F(\alpha ),\alpha =G(\beta )}$ ,

with polynomials ${\displaystyle F}$  and ${\displaystyle G}$  over ${\displaystyle K}$ . Now if ${\displaystyle Q}$  is the minimal polynomial for ${\displaystyle \beta }$  over ${\displaystyle K}$ , we can call ${\displaystyle Q}$  a Tschirnhaus transformation of ${\displaystyle P}$ .

Therefore the set of all Tschirnhaus transformations of an irreducible polynomial is to be described as running over all ways of changing ${\displaystyle P}$ , but leaving ${\displaystyle L}$  the same. This concept is used in reducing quintics to Bring–Jerrard form, for example. There is a connection with Galois theory, when ${\displaystyle L}$  is a Galois extension of ${\displaystyle K}$ . The Galois group may then be considered as all the Tschirnhaus transformations of ${\displaystyle P}$  to itself.

## History

In 1683, Ehrenfried Walther von Tschirnhaus published a method for rewriting a polynomial of degree ${\displaystyle n>2}$  such that the ${\displaystyle x^{n-1}}$  and ${\displaystyle x^{n-2}}$  terms have zero coefficients.

In his paper, Tschirnhaus referenced a method by Descartes to reduce a quadratic polynomial ${\displaystyle (n=2)}$  such that the ${\displaystyle x}$  term has zero coefficient.

In 1786, this work was expanded by E. S. Bring who showed that any generic quintic polynomial could be similarly reduced.

In 1834, G. B. Jerrard further expanded Tschirnhaus' work by showing a Tschirnhaus transformation may be used to eliminate the ${\displaystyle x^{n-1}}$ , ${\displaystyle x^{n-2}}$ , and ${\displaystyle x^{n-3}}$  for a general polynomial of degree ${\displaystyle n>3}$ .[3]