MaplePrimes - answers and comments on Question, solutions by radicals for polynomials ?

solve

Robert Israel — Tue, 01 Mar 2011 01:21:51 Z

For your solvable quintic, solve works:

solve(32*c^5-16*c^4-32*c^3+12*c^2+6*c-1, explicit);

For the sextic, it doesn't.

trig to radical...

acer — Tue, 01 Mar 2011 02:38:30 Z

Hey Axel, I see you're still musing over that usenet post. One might easily get Maple to convert the following roots of that quintic to radicals, by just calling `solve` with its Explicit option.

> seq(cos(i*Pi/11),i in [1,3,5,7,9]);

           /1    \     /3    \     /5    \      /4    \      /2    \
        cos|-- Pi|, cos|-- Pi|, cos|-- Pi|, -cos|-- Pi|, -cos|-- Pi|
           \11   /     \11   /     \11   /      \11   /      \11   /

But these roots of that sextic look like they are going to be a little tougher... ;)

seq(cos(i*Pi/13),i in [1,3,5,7,9,11]);

    /1    \     /3    \     /5    \      /6    \      /4    \  
 cos|-- Pi|, cos|-- Pi|, cos|-- Pi|, -cos|-- Pi|, -cos|-- Pi|, 
    \13   /     \13   /     \13   /      \13   /      \13   /  

       /2    \
   -cos|-- Pi|
       \13   /

acer

Boswell and Glasser

Robert Israel — Thu, 03 Mar 2011 00:51:24 Z

You might look at the paper of Boswell and Glasser "Solvable Sextic Equations", http://www.arxiv.com/abs/math-ph/0504001

They mention using Maple; I wonder if they have Maple code available.

Googling

hirnyk — Thu, 03 Mar 2011 01:23:29 Z

Also see the results of the "Solvable Sextic Equations" search in Google here.

The minimal polynomial of trigonometrics like cos (rational * Pi), sin(rational * Pi)

Axel Vogt — Fri, 11 Mar 2011 02:06:32 Z

Based on the given links and inputs at least I understand some parts.

For cos one basically one can use the Chebyshev polynomials, having the desired zeros (see for example stuff about integration methods and the needed nodes). Factoring them reduces to find an according factor (done numerically, may be interval arithmetics and 'shake' is more sound - but hairy to use and it would be more easy to increase working precision in case of troubles).

To special things have to obeyed:

In very low degree or special situations Maple spits out rationals or radicals ( cos(Pi/3), cos(Pi/4), ...), hence one needs to find the minimal polynomial of an algebraic number given as radical. Astonishingly I did not find usable codes for that and thus use ways provided by Maple. But I do not like PolynomialTools[MinimalPolynomial] (try it for sin(1) to understand why), instead gfun:-algfuntoalgeq seems more convenient.

The other thing is: for rationals beyond +-1 an automatical simplification invents signs to the original expression and this motivated me to write the code for linear expressions of the original task (ok, I also wanted to do some exercise in handling simple functional coding, since else I always omit it).

What I dislike: factorization is used ...

For the sinus there is a way to use the resultant. But that may introduce reducible solutions and one again would have to find and select factors. Instead enforce a conversion to the cosinus, which Maple would not be willing to choose. After that and embedding the core solution in a similar manner as above one can fall back to the solution that we already have.

trig_minimal_polynom.mws
trig_minimal_polynom.pdf

solve / explicit

Axel Vogt — Tue, 01 Mar 2011 01:44:41 Z

Thx ... I have to upgrade my version :-)

Do you see a way to use that option for the following (which
is the answer I got with Maple 12 and hesitate to re-code)?

1/2*RootOf(-1+3*_Z+3*_Z^2-4*_Z^3-_Z^4+_Z^5,index = 3)

Edited: I withdraw, allvalues does it ...