Coercion problem

[micha...@gmail.com]

I recently sumbled at following behaviour:
sage: P.<x> = PolynomialRing(QQ)
sage: w = x + 1
sage: w(oo)
+ Infinity
sage: v = 5
sage: v(oo)
A positive finite number

This behaviour is strange (altough works as desined). For nonconstant polynomial it has the same effect as calculating limit, for constant polynomial it gives (seemingly) non-sensical result. Could the default behaviour be changed so in case of constant polynomial instead of coercing operands it would return simply the value of the polynomial (5 in this example)?

Regards,
MG

[Robert Bradshaw]

I assume you meant

sage: v = P(5)

sage: v(oo)
A positive finite number

This is because the elements of QQ coerce to the parent of oo, which
is the "signed infinity ring." This is so we have

sage: P.<x> = PolynomialRing(QQ)

sage: w = x + 5
sage: v = w - x
w(1.0)
6.00000000000000
sage: v(1.0)
5.00000000000000
sage: parent(w(1.0)) is parent(v(1.0))
True

I suppose now that we have pushouts we could let the result lie in the
(affine) extension of the rationals. Manipulation with the unsigned
infinity would yield the projective extension. There is the open
question of what the parent of oo should be though. Some kind of
affine extension of ZZ?

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