Thanks, Simon, for the explanation of coercions. > > I encountered some weird behavior in LaurentPolynomialRing, such as > > the nontransitivity of the '==' relation. > > Note that it is virtually impossible to have a transitive ==-relation > that at the same time provides certain mathematical features that are > wanted by most people.
However if all coercion maps involved are injective then can't I expect == to be preserved? > > Also, factoring in the fraction field caused an error. > > It is implemented by factoring numerator and denominator, see below; > the problem is in factoring Laurent series. It fails in factoring the unit element of S. That is pretty strange to me. > > If R is the polynomial ring underlying S > > then there should be a canonical isomorphism installed between F and > > the fraction field K > > of R. > > There *is* a coercion from K to F, but of course not in the opposite > direction: Yeah, this unimpeachable coercion comes from the universal mapping property of fraction fields. > sage: S.<x> = LaurentPolynomialRing(QQ) > sage: F = S.fraction_field() > sage: F > Fraction Field of Univariate Laurent Polynomial Ring in x over > Rational Field > sage: R = S.polynomial_ring() > sage: K = R.fraction_field() > sage: F.has_coerce_map_from(K) > True > Note that there is a coercion from S to F which is injective > and also Sage can not find the pushout: > sage: pushout(S, FractionField(R)) > Traceback (most recent call last): > ... > CoercionException: ... Since the coercion from S to FractionField(R) is injective, the pushout must necessarily be FractionField(R). On the other hand, FractionField(R) does not know about membership in the subring S. > > sage: (1/S.gen(0)).factor() > > Traceback (most recent call last) > > ... > > AttributeError: 'sage.rings.integer.Integer' object has no attribute > > 'dict' s = 1/S.gen(0) sn = s.numerator() sn.factor() <boom> > The last line will show you the source code, and you will see that it > is attempted to factor numerator and denominator. As it turns out, the > problem lies in the factorisation of Laurent series. The problem occurs when the Laurent polynomial ring unit element is factored! --Mark -- To post to this group, send an email to sage-devel@googlegroups.com To unsubscribe from this group, send an email to sage-devel+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/sage-devel URL: http://www.sagemath.org
