# Difficulties with resultants and Tschirnhaus transformations

I'm experimenting with some polynomial equations, and as a test I'm working through the example given here:

http://www.oocities.org/titus_piezas/...

Basically, in order to eliminate the first two terms of the quintic $x^5-x^4-x^3-x^2-x-1$ we need to find a transformation $y=x^2+ax+b$ (such a polynomial transformation is called a Tschirnhaus transformation, after its first discoverer), for which $a$ and $b$ have the effect of producing a quintic equation in $y$ but without $y^4$ or $y^3$ terms. This is as far as I've got so far:

R.<a,b> = QQ[]
S.<y> = R[]
T.<x> = S[]
p = x^5-x^4-x^3-x^2-x-1
res = p.resultant(y-x^2-a*x-b)
rc = res.coefficients()
solve([rc[-2],rc[-3]],[a,b])

TypeError: a is not a valid variable.


Drat! So I have two questions: why is a not a valid variable, and why can I not isolate individual coefficients of res? I can obtain all the coefficients of res, but I can't isolate just one, with something like res.coeff(y,3).

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If I declare variables, then solve also works

    sage: a, b = var('a b')
sage: solve([rc[-2],rc[-3]],[a,b])
[[a == (-11/7), b == (-2/7)], [a == -3, b == 0]]


Sort by » oldest newest most voted As this is a purely algebraic problem, you can avoid the symbolic solve as follows:

sage: I = R.ideal([rc[-2],rc[-3]])
sage: I.variety()
[{a: -11/7, b: -2/7}, {a: -3, b: 0}]

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Could you please accept the answer if it suits you ? (on the left)

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