| Version 6 |
Version 5 |
| A polynomial transformation which transforms a polynomial to another with certain zero-coefficients is called a |
A polynomial transformation which transforms a polynomial to another with certain zero-coefficients is called a |
| \emph{Tschirnhaus Transformation}. It is thus an invertible transformation of the form $x \mapsto g(x)/h(x)$ where $g,h$ are polynomials over the base field $K$ (or some subfield of the splitting field of the polynomial being transformed). If $\gcd(D(x),f(x)) = 1$ then the Tschirnhaus transformation becomes a polynomial transformation mod f. |
\emph{Tschirnhaus Transformation}. It is thus an invertible transformation of the form $x \mapsto g(x)/h(x)$ where $g,h$ are polynomials over the base field $K$ (or some subfield of the splitting field of the polynomial being transformed). If $\gcd(D(x),f(x)) = 1$ then the Tschirnhaus transformation becomes a polynomial transformation mod f. |
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| Specifically, it concerns a substitution that reduces finding the roots of the polynomial |
Specifically, it concerns a substitution that reduces finding the roots of the polynomial |
| $$ |
|
| \textmd{p} = T^n + a_1T^{n-1} + ... + a_n = \prod_{i=1}^n |
\textmd{p} = T^n + a_1T^{n-1} + ... + a_n = \prod_{i=1}^n |
| (T-r_i)\in k[T] |
(T-r_i)\in k[T] |
| $$ |
|
| to finding the roots of another \textmd{q} - with less parameters |
to finding the roots of another \textmd{q} - with less parameters |
| - and solving an auxiliary polynomial equation \textmd{s}, with |
- and solving an auxiliary polynomial equation \textmd{s}, with |
| $\deg(s)<\deg(p \cap q).$ |
$\deg(s)<\deg(p \cap q).$ |
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| Historically, the transformation was applied to reduce the general quintic equation, to simpler resolvents. Examples due to Hermite and Klein are |
Historically, the transformation was applied to reduce the general quintic equation, to simpler resolvents. Examples due to Hermite and Klein are |
| respectively: The principal resolvent |
respectively: The principal resolvent |
| $$ |
|
| K(X):=X^5+a_0X^2+a_1X+a_3 |
K(X):=X^5+a_0X^2+a_1X+a_3 |
| $$ |
|
| and the Bring-Jerrard form |
and the Bring-Jerrard form |
| $$ |
|
| K(X):=X^5+a_1X+a_2 |
K(X):=X^5+a_1X+a_2 |
| $$ |
|
| Tschirnhaus transformations are also used when computing Galois |
Tschirnhaus transformations are also used when computing Galois |
| groups to remove repeated roots in resolvent polynomials. Almost any transformation will work but it is |
groups to remove repeated roots in resolvent polynomials. Almost any transformation will work but it is |
| extremely hard to find an efficient algorithm that can be proved |
extremely hard to find an efficient algorithm that can be proved |
| to work. |
to work. |
