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Difficulties with resultants and Tschirnhaus transformations

asked 2014-09-21 11:31:10 +0200

Alasdair gravatar image

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

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() 

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]]

SL gravatar imageSL ( 2014-09-22 19:04:32 +0200 )edit

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answered 2014-09-21 11:44:25 +0200

FrédéricC gravatar image

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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Perfect - simple and elegant. Thanks very much!

Alasdair gravatar imageAlasdair ( 2014-09-21 16:45:26 +0200 )edit

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

FrédéricC gravatar imageFrédéricC ( 2014-09-21 16:50:52 +0200 )edit

@Alasdair: when you are satisfied with an answer, you can formally accept it by clicking the tick sign on the top left of the answer, below the answer's score. Additionally, you can upvote any answer that helped you, using the upward arrow on top of the answer's score. Accepting the answer marks your question as satisfactorily answered in the list of questions on the main page (helping volunteers to focus on unsolved questions). Additionally, when you accept an answer, you get 2 karma points and the answerer gets 25 karma points. When you upvote, the answerer gets 10 karma points.

slelievre gravatar imageslelievre ( 2014-09-21 18:37:35 +0200 )edit

Thanks for the reminder: I've accepted the answer.

Alasdair gravatar imageAlasdair ( 2014-09-22 02:21:30 +0200 )edit

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Asked: 2014-09-21 11:31:10 +0200

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Last updated: Sep 21 '14