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Difference between revisions of "Tschirnhausen transformation"

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''Tschirnhaus transformation''
 
''Tschirnhaus transformation''
  
A transformation of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120180/t1201801.png" />th degree polynomial equation
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A transformation of an $n$th degree polynomial equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120180/t1201802.png" /></td> </tr></table>
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$$f(X)=X^n+a_{n-1}X^{n-1}+\ldots+a_0=0$$
  
 
by a substitution of the form
 
by a substitution of the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120180/t1201803.png" /></td> </tr></table>
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$$Y=\alpha_0+\alpha_1X+\ldots+\alpha_{n-1}X^{n-1}$$
  
 
to an equation
 
to an equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120180/t1201804.png" /></td> </tr></table>
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$$Y^n+b_{n-1}Y^{n-1}+\ldots+b_0=0,$$
  
hopefully of simpler form. For instance, the general equation of degree five can be brought to the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120180/t1201805.png" /> (the so-called Bring–Jerrard normal form) using only quadratic roots. Quite generally, the terms of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120180/t1201806.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120180/t1201807.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120180/t1201808.png" /> can always be eliminated by a suitable Tschirnhausen transformation.
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hopefully of simpler form. For instance, the general equation of degree five can be brought to the form $Y^5+Y+b=0$ (the so-called Bring–Jerrard normal form) using only quadratic roots. Quite generally, the terms of degree $n-1$, $n-2$, $n-3$ can always be eliminated by a suitable Tschirnhausen transformation.
  
 
The procedure is named after Count E.W. von Tschirnhaus, who described these transformations in [[#References|[a1]]]. Contrary to the beliefs of Tschirnhaus and Jerrard at that time (around 1683), these transformations do not help solving general polynomial equations of degree larger than four (see also [[Galois theory|Galois theory]]).
 
The procedure is named after Count E.W. von Tschirnhaus, who described these transformations in [[#References|[a1]]]. Contrary to the beliefs of Tschirnhaus and Jerrard at that time (around 1683), these transformations do not help solving general polynomial equations of degree larger than four (see also [[Galois theory|Galois theory]]).

Revision as of 17:22, 23 September 2014

Tschirnhaus transformation

A transformation of an $n$th degree polynomial equation

$$f(X)=X^n+a_{n-1}X^{n-1}+\ldots+a_0=0$$

by a substitution of the form

$$Y=\alpha_0+\alpha_1X+\ldots+\alpha_{n-1}X^{n-1}$$

to an equation

$$Y^n+b_{n-1}Y^{n-1}+\ldots+b_0=0,$$

hopefully of simpler form. For instance, the general equation of degree five can be brought to the form $Y^5+Y+b=0$ (the so-called Bring–Jerrard normal form) using only quadratic roots. Quite generally, the terms of degree $n-1$, $n-2$, $n-3$ can always be eliminated by a suitable Tschirnhausen transformation.

The procedure is named after Count E.W. von Tschirnhaus, who described these transformations in [a1]. Contrary to the beliefs of Tschirnhaus and Jerrard at that time (around 1683), these transformations do not help solving general polynomial equations of degree larger than four (see also Galois theory).

A generalization of the Tschirnhausen transformation plays a role in the original proof of the Abhyankar–Moh theorem.

References

[a1] E.W. von Tschirnhaus, Acta Eruditorium (1683)
[a2] H. Weber, "Lehrbuch der Algebra" , I , Chelsea, reprint pp. Chap. 6 (First ed.: 1898)
[a3] A.L. Cayley, "On Tschirnhausen's transformation" Philos. Trans. R. Soc. London , 151 (1861) pp. 561–578
How to Cite This Entry:
Tschirnhausen transformation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tschirnhausen_transformation&oldid=16919
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article