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'Tchirnhaus transformations'
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| Title of object: |
Tchirnhaus transformations |
| Canonical Name: |
TchirnhausTransformations |
| Type: |
Definition |
| Created on: |
2003-08-09 19:52:24 |
| Modified on: |
2003-09-29 23:28:18 |
| Keywords: |
reduction, polynomial, resultant |
| Synonyms: |
Tchirnhaus transformations=Tschirnhausen Transformation |
Revision comment (for changes between this and next version):
| Changes for correction #2362 ('two typos'). |
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Content:
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 thepolynomial being transformed). If $\gcd(D(x),f(x)) = 1$ then the Tschirnhaus transformation becomes a polynomial transformation mod f.
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
(T-r_i)\in k[T]
to finding the roots of another \textmd{q} - with less parameters
- and solving an auxiliary polynomial equation \textmd{s}, with
$\deg(s)<\deg(p \cap q).$
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
K(X):=X^5+a_0X^2+a_1X+a_3
and the Bring-Jerrard form
K(X):=X^5+a_1X+a_2
Tschirnhaus transformations are also used when computing Galois
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
to work. |
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