On Sep 24, 6:56 pm, kcrisman <kcris...@gmail.com> wrote:
> If I make a polynomial ring using
>
> sage: b = PolynomialRing(ZZ, 'x')
>
> I get some odd behavior. Namely,
>
> sage: bool(b(x)==x)
> True
I personally find this slightly worrisome, but this was a design
decision (equality respects automatic coercions)
> sage: b(x)
> x
> sage: type(b(x))
> <something about element of the ring>
> sage: type(x)
> <symbolic expression>
>
> This isn't really that odd, but still I don't know whether it is good
> that one can still use x as a symbolic variable.
This is the only possibility, because the "var('x')" command executed
by default at startup did the assignment
x = SR('x')
and you haven't bound x to any other object. Once you execute
x = b.0
[ or one of its implicit forms like b.<x>=PolynomialRing(ZZ,'x')] then
x is no longer referencing the "symbolic expression x", but the
"univariate polynomial x" instead.
> sage: a = FractionField(PolynomialRing(ZZ, 'x'))
> sage: a(1/x)
> <weird error that seems to imply it has not coerced x to the
> polynomial ring>
sage: Xsym = var('x')
sage: P=PolynomialRing(ZZ,'x')
sage: Xpoly = P.gen(0)
sage: FP=FractionField(P)
sage: Xratfunc = FP.gen(0)
sage: bool(1/Xratfunc == 1/Xsym)
True
sage: FP(1/Xsym)
TypeError: denominator must be a unit
It seems things are mainly going in the other direction:
sage: cm = sage.structure.element.get_coercion_model()
sage: cm.explain(parent(Xsym),parent(Xratfunc),operator.eq)
Coercion on right operand via
Conversion via _symbolic_ method map:
From: Fraction Field of Univariate Polynomial Ring in x over
Integer Ring
To: Symbolic Ring
Arithmetic performed after coercions.
Result lives in Symbolic Ring
Symbolic Ring
which is correct: A rational function with integer coordinates can
definitely be coerced into the symbolic ring, but in the other
direction, one can only hope for a conversion map that is partially
defined:
sage: FP(1/SR('y'))
TypeError: y is not a variable of Univariate Polynomial Ring in x over
Integer Ring
It looks to me as if FP(1/Xsym) is trying to convert via P:
sage: P(1/Xsym)
TypeError: denominator must be a unit
Given that the error returned for FP(1/Xsym) is nonsensical, it's
reasonable to call it a bug. The obvious way to fix it is by handling
conversions from SR a little more intelligently. Beware though,
because the parents could be really nasty:
sage:
F=PolynomialRing(FractionField(PolynomialRing(FractionField(PolynomialRing(ZZ,'x')),'y')),'z')
sage: F(SR( '1/x+1/y*x'))
sage: F(1/Xsym)
which, surprisingly, works already!! So yes, it's a bug and it's
probably something simple that's missing on the optimized univariate
polynomial ring over the integers.
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