A few questions about solve and exact polynomial roots

Mike Valenzuela on  Dec 2,14:24:50 wrote:

>Hello all,
>I was messing around with some eigenvectors and values from a simple system
>and came across the *casus irreduciblis* problem: sometimes complex
>radicals are needed to express real solutions. However, I've been trying to
>reduce the solution to real non-algebraic solutions.
>(specified here)
>
http://en.wikipedia.org/wiki/Casus_irreducibilis#Non-algebraic_solution_in_terms_of_real_quantities
>(and here)
>
http://en.wikipedia.org/wiki/Cubic_function#Trigonometric_.28and_hyperbolic.29_method

>Anyways, I am wondering if maxima natively supports
>(1) Returning non-algebraic solutions from its solving system, especially
>if they are simplier.
>(2) Some combination of trig simplification rules to recover the simple
>form given in the above link.

>I suppose users could manually use the above identities when applicable,
>but they would have to be aware of it.



>From my "odes" package (to appear in next Maxima relase)
function   solvet(eq,x)
returns rectform solution of polynomial equation.
In "casus irreducibilis" give real solutions expressed
in trigonometric functions.

 Examples:
(%i2) solvet(x^3-3*x^2+1,x);
(%o2) [x=2*cos(%pi/9)+1,x=2*cos((5*%pi)/9)+1,x=2*cos((7*%pi)/9)+1]

(%i3) solvet(x^6-3*x^5-3*x^4+12*x^3-3*x^2-6*x+2=0,x);
(%o3)
[x=1,x=1-sqrt(3),x=sqrt(3)+1,x=2*cos((2*%pi)/9),x=2*cos((4*%pi)/9),x=2*cos((8*%pi)/9)]

(%i4) solvet(x^3-15*x-5,x);
(%o4)
[x=2*sqrt(5)*cos(atan(sqrt(19))/3-(2*%pi)/3),x=2*sqrt(5)*cos(atan(sqrt(19))/3+(2*%pi)/3),x=2*sqrt(5)*cos(atan(sqrt(19))/3)]

(%i5) solve(x^3-15*x-5,x);
(%o5)
[x=(-(sqrt(3)*%i)/2-1/2)*((5*sqrt(19)*%i)/2+5/2)^(1/3)+(5*((sqrt(3)*%i)/2-1/2))/((5*sqrt(19)*%i)/2+5/2)^(1/3),x=((sqrt(3)*%i)/2-1/2)*((5*sqrt(19)*%i)/2+5/2)^(1/3)+(5*(-(sqrt(3)*%i)/2-1/2))/((5*sqrt(19)*%i)/2+5/2)^(1/3),x=((5*sqrt(19)*%i)/2+5/2)^(1/3)+5/((5*sqrt(19)*%i)/2+5/2)^(1/3)]

best

Aleksas D