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Re: How to get an answer as a Root object?
In article <d6mlqh$g9o$1 at smc.vnet.net>,
Andrzej Kozlowski <akoz at mimuw.edu.pl> wrote:
> Mathod 1 is of course classical (discussed for example in 1991 Henri
> Cohen's "A Course in Computational Algebraic Number Theory" on page
> 100)
Thanks for the ref. I've used this idea regularly since I first
encountered the Recognize package but have not seen it in the literature
(though I knew it would be classical).
> but I don't like the fact that you have to (or at least it seems
> to me you have to) guess the degree of the polynomial used in
> Recognize.
You can increment the degree and determine the minimal degree (using
FixedPoint or somesuch).
expr = (-1 + Sqrt[2] - Sqrt[(1 + 2 Sqrt[2] - s) (-1 + s)] + s) *
(-1 - Sqrt[2] + Sqrt[2] s -
Sqrt[-1 - 2 Sqrt[2] + 2 (2 + Sqrt[2] - s) s]) -
(-1 - Sqrt[2] + Sqrt[2] s + Sqrt[-1 - 2 Sqrt[2] +
2 (2 + Sqrt[2] - s) s]) (-2 - 2 Sqrt[2] + 2 s +
2 Sqrt[1 - (1/4) (-1 - Sqrt[2] +
Sqrt[(1 + 2 Sqrt[2] - s) (-1 + s)] + s)^2]);
<< NumberTheory`
x = Chop[s /. FindRoot[expr == 0, {s, 2.5}, WorkingPrecision -> 100]]
Table[{n, Last[FactorList[Recognize[x, n, t]]]}, {n, 10, 20}]
Also, you can verify the solution by back-substitution:
RootReduce[expr /. s -> Root[Function[t,
256 t^12 - 3072 t^11 + 15872 t^10 - 46080 t^9 + 81425 t^8 -
86152 t^7 + 43568 t^6 + 10832 t^5 - 32136 t^4 + 21024 t^3 -
6112 t^2 + 512 t + 64], 8]]
Cheers,
Paul
--
Paul Abbott Phone: +61 8 6488 2734
School of Physics, M013 Fax: +61 8 6488 1014
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