On Fri, Mar 07, 2025 at 04:47:04PM +0100, 'Ralf Hemmecke' via FriCAS - computer
algebra system wrote:
> Dear Waldek,
>
> thanks for the reply and apologies for being imprecise.
>
> Actually the algebraic numbers I talk about are rational expression of
> nested radicals. In fact, I get all roots of a polynomial via radicalRoots $
> RadicalSolvePackage(ZZ). Degree is <= 4 (mostly 2).
> These roots undergo some rational operation and multiplication with other
> roots. But let's forget about the latter. I am actually talking about all
> roots.
>
> I know that I can find rational intervals for all real roots as tight as I
> want them.
> I basically need a matching of these intervals with the respective radical
> expressions that I get from radicalRoots.
You should understand the in generic case there is a single multivalued
root: depending on branches of radicals you get all root of corresponding
polynomial. AFAIK the only _general_ method to track the roots is
evaluation via some local field. I do not think there is any regularity
in results from different local fiels, so practically one have to
choose one and stick to it. In a sense most typical local field
is that of complex numbers and practical methods of doing
evaluation are numeric.
AFAICS you are interested in _very_ special cases. It is quite
likely that you can do what you want by case analysis. For
example you can handle square roots of rationals by evaluating
exactly number under square root and looking at its sign.
In RootUtilities we handle bunch of cases of for equations of
degree 3 and 4. But already for degree 3 there is case of
3 real roots, each having complex expression.
> Additionally, given an algebraic number z, it would be great if FriCAS had a
> way to find a p \in Z[x] such that p(z)=0 (minimal poly for z).
> I looked at definingPolynomial and it is not so helpful, as I would have to
> construct the minimal polynomial over QQ myself.
Yes, it would be nice to have this, but it is currently unimplemented.
--
Waldek Hebisch
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