Re: Real Algebraic Numbers

[Bill Allombert <Bill.Allombert@math.u-bordeaux.fr>]

On Wed, May 20, 2020 at 10:41:06PM +0200, Gereon Kremer wrote:
> Hi all,
> 
> I'm trying to use PARI for computations with real algebraic numbers,
> ultimately aiming for a CAD implementation. In particular, I'd need the
> following operations:
> 
> 1) Multivariate resultants. Does the resultant() method from 8.10.1 work
> on multivariate polynomials (in a main variable)?

In principle yes, in practice it depends on the number of variables.

> 2) Isolate real roots from a univariate polynomial. I got realroots() to
> work, however I'm not sure whether there is a built-in that directly
> couples the numeric approximation with the defining polynomial to form a
> number type. I guess number fields go in this direction to some degree,
> is this the way to go?

Yes.

> 3) Isolate real roots from a multivariate polynomial and a real
> algebraic assignment (for all but one variable from the polynomial).
> 
> Can anyone give me some hints how to do this with PARI? (or,
> alternatively, tell me that I should not use PARI for that and maybe
> even suggest an alternative...)

I would say it depends on the number of variables you need.
PARI/GP is good for one or two variables but is not suited for 
large number of variables.
On the other hand it can handle large algebraic degrees.

Cheers,
Bill.