Re: Real Algebraic Numbers

[Karim Belabas <Karim.Belabas@math.u-bordeaux.fr>]

* Gereon Kremer [2020-05-20 22:41]:
> 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)?

It works, but it will get much worse as the number of variable increases. I
found about 10 variables to be a practical limit for many computations of this
kind even for moderate sizes (e.g., determinant of a 10 x 10 matrix): we only
have dense representations for polynomials and the number of possible monomials
explodes quickly (even if the corresponding coefficient is 0).

> 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?
>
> 3) Isolate real roots from a multivariate polynomial and a real
> algebraic assignment (for all but one variable from the polynomial).

No built-in way. With number fields, you can specify real places attached to
the real roots of your defining polynomial and use functions like nfeltsign or
nfpolsturm (which use algebraic methods and should give a correct result
independent of realprecision)

> Can anyone give me some hints how to do this with PARI? 

Can you post a minimal example ?

Cheers,

    K.B.
--
Karim Belabas, IMB (UMR 5251)  Tel: (+33) (0)5 40 00 26 17
Universite de Bordeaux         Fax: (+33) (0)5 40 00 21 23
351, cours de la Liberation    http://www.math.u-bordeaux.fr/~kbelabas/
F-33405 Talence (France)       http://pari.math.u-bordeaux.fr/  [PARI/GP]
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