[Axiom-developer] Re: [sage-devel] Re: MAGMA question

[William Stein]

On Wed, 18 Oct 2006 13:05:52 -0700, Page, Bill <address@hidden>  
wrote:
> > Incidentally, SAGE already has a useful shorthand for the
> > above:
> >
> > sage: x = polygen(ZZ)
> > sage: parent(x)
> > Univariate Polynomial Ring in x over Integer Ring
>
> I think you need to define the semantics very carefully here.
> As an Axiom user I want to interpret the Sage expressions
> above in a slightly different language. What you mean is that
> x is now a member of a certain class, i.e. it's "parent" and
> polygen is a generator of members of this class. Here the
> value is explicit by the type of the result, i.e. parent, is
> implicit. This is sort of the opposite of the convention used
> in Axiom.
>
> In Sage how would I create a symbol whose value is the Integer
> 1 considered as a Polynomial? E.g. (5) above.

sage: R = ZZ['x']
sage: a = R(1)
sage: type(a)
<class 'sage.rings.polynomial_element.Polynomial_integer_dense'>
sage: parent(a)
Univariate Polynomial Ring in x over Integer Ring
sage: a
1

You could make 1 in any ring R the same way.

> > Of course, in SAGE you can use PARI polynomials:
> >
> > sage: x = pari('x')
> > sage: y = pari('y')
> > sage: f = x*(y^2+7/3)
> > sage: f^3
> > (y^6 + 7*y^4 + 49/3*y^2 + 343/27)*x^3
>
> And now you can do the same think using Axiom :-)

And now the interface is officially released...

> address@hidden page]$ sage
> --------------------------------------------------------
> | SAGE Version 1.4, Build Date: 2006-10-05             |
> | Distributed under the GNU General Public License V2. |
> --------------------------------------------------------
>
> experimental
>
> sage: x = axiom('x')
> sage: y = axiom('y')
> sage: f = x*(y^2+7/3)
> sage: f^3
>
>    3 6     3 4   49  3 2   343  3
>   x y  + 7x y  + -- x y  + --- x
>                   3         27
> Type: Polynomial Fraction Integer
>
> sage:
>
> > > Anyhow, it is just one pet peeve in MAGMA which seems
> > > easy enough to relieve somehow.
> >
> > What do you think of
> >      x = polygen(ZZ)      for x in ZZ[x]
> > and
> >      x = polygen()        for a formal x (whose properties
> > are yet to be determined)
>
> What would be the value of parent(x) for the second case?

We haven't decided yet --  I'm thinking something like
    Formal Univariate Polynomial Ring.

William