Tschirnhausen transformation

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