[GiNaC-list] Polynomial Zeros

[Dima Pasechnik]

On Wed, Oct 11, 2023 at 08:14:40AM +0000, Vladimir V. Kisil wrote:
> >>>>> On Mon, 9 Oct 2023 15:05:28 -1000, Charles White <whitece6 at hawaii.edu> said:
> 
>     CW> Hi GiNaC!  Is there a built-in method to find all zeros of a
>     CW> polynomial? I'm trying to find symbolic zeros if possible and
>     CW> numeric zeros for all other cases.
> 
>     If I remember correctly, analytic expression for polynomial roots is
>   possible only up to the four order (inclusively). These formulae can
>   be directly coded by a user.

The situation is actually more subtle - if you want as many symbolic roots as there are.
One can compute the Galois group
of the polynomial (not in GiNaC, in a computer algebra system like pari/GP),
and its soluble factors correspond to exact symbolic
roots. Such computations need to use info about permutation groups of
the degree equal to the degree of the polynomial, and become practically
infeasible for degrees bigger than 20 or so.

For a polynomial with a soluble Galois group (a factor of your original
polynomial), writing roots in radicals is possible.
Software I know for this is a GAP package: https://www.gap-system.org/Packages/radiroot.html
(which calls pari/GP for the heavy number-theoretic lifting)

Practically speaking (for software in C/C++), one can call pari/GP as a
dynamic library (or just as a shell script) to do the factoring; 
small degree (at most 4) factors will automatically give symbolic roots,
bigger degree factors might be analysed further by means of its Galois
groups.

HTH
Dima

> For a generic polynomial with integer
>   coefficient factor() method may give hint on possible integer
>   roots. Numerical approximation may be obtained by fsolve(), see
>   section "2.2 What it can do for you" of the tutorial.  
> -- 
> Vladimir V. Kisil                 http://www1.maths.leeds.ac.uk/~kisilv/
>   Book:      Geometry of Mobius Maps       https://doi.org/10.1142/p835
>   Soft:      Geometry of cycles         http://moebinv.sourceforge.net/
>   Jupyter notebooks:        https://github.com/vvkisil?tab=repositories
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