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Re: polynomial of 7th order, some structure

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  • Subject: [mg100751] Re: polynomial of 7th order, some structure
  • From: Alexei Boulbitch <Alexei.Boulbitch at>
  • Date: Sat, 13 Jun 2009 06:01:14 -0400 (EDT)

hi, I have an equation i have been trying to solve, but no amount of
Solve, Reduce or ToRadical seems to produce an answer.

x(1+x^2)^3 == a x^4 +b x^3 +...+e

I know that it has one or three real roots, depending on combinations
of coefficients, with only one root at x>0.  That is the one that I
want, the root that is > 0.
Does anyone have any ideas what I can do?

I can plot the Root[] results, and I can see that it is usually the
same one root from the list, for most coefficient combinations.

Any way i can get the formula without plugging in the values?

Hi, Tolia,

your question is not about Mathematica, but about mathematics. My excuses to moderator.
Are n't you aware of the Galois theorem stating 
that algebraic equation of a general form only then have solution in radicals (that is what you are looking for) 
when its power is below or equal to 5? You have equation of the power 7 with 5 arbitrary constants, 
do you hope that it may belong to exceptions of this theorem? Then you should have good reasons for that, and these
reasons should most probably go beyond the form of the equation itself. For example, they may be in the physical
(chemical, etc.) nature of the phenomenon that gave rise to this equation. I met such situations.

If I am right and no such reasons exist, the only thing you can reasonably look for in this case may be conditions 
imposed on the coefficients a, b ... at which such a (positive) root appears and disappears. You may also look 
for an approximate solution for x>0 in the close vicinity of such a bifurcation point. But this again more the
mathematical question, of how to look for such a bifurcation point. If it is formulated correctly, simple 
Mathematica tools will be applicable.

Good lack, Alexei

Alexei Boulbitch, Dr., habil.
Senior Scientist

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