From: mckay@alcor.concordia.ca (John McKay)
Newsgroups: sci.math
Subject: Re: tschirnhaus transformations
Date: 29 Sep 1996 07:49:57 GMT

My definition of a Tschirnhaus transformation is:

An invertible transformation of the form   x -> N(x)/D(x)
where N,D are polynomials over K (usually taken to be the
base field but may be some subfield of the splitting field
of the polynomial being transformed).
If gcd(D(x),f(x)) = 1 then the T.t. becomes a polynomial
transformation mod f.
 Tschirnhaus transformations are used when computing Galois
groups to remove repeated roots in resolvent polynomials.
It is "strange but true" that almost any T.t. will work but
it is extremely hard to find an (efficient) algorithm that
can be proved to work.
Tschirnhaus was not a mathematician. The best-known T.t. is
x -> x - a[1]/n to replace x^n+a[1]x^(n-1)+...+a[n] by
x^n+a'[2]x^(n-2)+...+a'[n] but his idea was to use a
polynomial transformation of degree n-1 to reduce to
x^n+c and thus show all polynomials to be solvable!






 In article <324D9266.6E47@student.utwente.nl>,
Wilbert Dijkhof  <s9400842@student.utwente.nl> wrote:
>I would like to know what the Tschirnhaus and BringJerrard transformations
>are? These transformations reduce the terms in polynomials. 
>Example, what transformation does the following:
>x^5+ax^4+bx^3+cx^2+dx+e=0 => z^5+pz+q=0 with x=f(z) is a polynomial in z.
>
>Thanks, Wilbert


-- 
Deep ideas are simple.
                      Odd groups are even.
                                          Even simples are not;
and Gal/F2(t)(x^24-x-t)  = Mathieu group, M24.