On Thu, May 22, 2008 at 7:03 AM, Simon King <[EMAIL PROTECTED]> wrote:
>
> Dear Marc,
>
> let me try some explanations.
>
> On May 22, 1:43 pm, Marc Roeder <[EMAIL PROTECTED]> wrote:
>> sage: QX=MPolynomialRing(QQ,2,'xy')
>> sage: x in QX # no variables assinged to indeterminates yet...
>> False
>
> If you start Sage, x is already defined:
> sage: type(x)
> <class 'sage.calculus.calculus.SymbolicVariable'>
>
> The apparent reason is that 'x' is a typical name for a symbolic
> variable. However, when one isn't aware that x is pre-defined, strange
> things may happen.
>
> Now, defining
> sage: QX=MPolynomialRing(QQ,2,'xy')
> does *not* change x!
>
> Sage makes a clear distinction between the default symbolic variable
> with identifier 'x' and the ring variable with the name 'x'. The ring
> variable can be accessed with
> sage: QX.gen(0)
> x
> sage: x is QX.gen(0)
> False
>
>> sage: indets=QX.objgens()[1];indets
>> (x, y)
>> sage: x in indets
>> True
>> But x and indets[0] have different types.
>
> You are right:
> sage: type(QX.objgens()[1][0])
> <type
> 'sage.rings.polynomial.multi_polynomial_libsingular.MPolynomial_libsingular'>
> sage: type(x)
> <class 'sage.calculus.calculus.SymbolicVariable'>
> sage: x in QX.objgens()[1]
> True
>
> I think this is a confusing behaviour that probably has the following
> reason:
> sage: x==QX.gen(0)
> x == x
>
> This is a symbolic equation (since x is symbolic) that evaluates to
> True, and consequently you get the wrong answer that the symbolic
> variable x belongs to indets.
NO. That is not what is going on. The explanation is that there is
a canonical map from QX to the symbolic ring. The in command
doesn't get confused by x == QX.gen(0) being a symbolic equality,
since it calls bool on that.
>
> In x==QX.gen(0) apparently the two *different* objects with coinciding
> names got mixed up. Do the more experienced Sage users agree with me
> that this is not nice (or a bug)?
I definitely do not agree with you.
>
>> 2. indeterminates are sometimes shared between rings:
>> sage: R.<x,y>=QQ[];R
>
> By the way, if you use that syntax, the symbolic variable x is
> overwritten, and now x is known to Sage as a ring generator:
> sage: R.<x,y>=QQ[]
> sage: type(x)
> <type
> 'sage.rings.polynomial.multi_polynomial_libsingular.MPolynomial_libsingular'>
>
>> sage: R2=MPolynomialRing(QQ,2,'xy')
>> sage: R==R2
>> True
>
> Yes, they are the same, in the sense that they have the same base ring
> and the same variable names in the same *order*.
> Compare
> sage: R3=MPolynomialRing(QQ,2,'yx')
> sage: R==R3
> False
>
> Even more, R and R2 are not only the same but *identical*:
> sage: R is R2
> True
>
> Therefore, x (which is now defined as a generator of R) is also
> generator of R2 (since R2 is identical with R2), but it is not a
> generator of R3:
> sage: x in R3
> False
>
>> sage: R2=MPolynomialRing(QQ,1,'x')
>> sage: x in R2
>> False
>
> Yes, in this example R2 is not identical with R, and by consequence x
> does not belong to R2.
>
> In conclusion, i believe that
> 1) pre-defining x may be confusing, but this is Sage tradition
It makes a lot of things much more concise. Sage is to be *used*,
not to be some hard-to-use abstract formal system.
> 2) sage: bool(var('x')=='x')
> True
> is a bug, IMO.
I disagree. There is a canonical coercion to the symbolic ring.
> 3) your observations concerning generators of R,R2 can be explained by
> the distinction between "the same" and "identical".
Yes.
>
> >
>
--
William Stein
Associate Professor of Mathematics
University of Washington
http://wstein.org
--~--~---------~--~----~------------~-------~--~----~
To post to this group, send email to sage-support@googlegroups.com
To unsubscribe from this group, send email to [EMAIL PROTECTED]
For more options, visit this group at
http://groups.google.com/group/sage-support
URLs: http://www.sagemath.org
-~----------~----~----~----~------~----~------~--~---
