Hi Kwankyu, On 24 Sep., 13:36, Kwankyu Lee <ekwan...@gmail.com> wrote: > I think one should not take it for granted a polynomial ring is a graded > ring.
No. Of course it is granted! > Of course, you can view a polynomial ring as a graded ring. But one > work with a polynomial ring, perhaps more often, not as a graded ring. Sure, nobody forces you to use the .degree() or .degrees() method. > With a polynomial in a polynomial ring not as a graded ring, you have > methods degree(x), degrees(), total_degree(). Hang on. If you *DO* use these methods, then you *ARE* working in a graded ring. > In this context, degree() can > be most naturally identified with total_degree(), that is the sum of the > exponents of variables. Certainly that grading should be the default. As I already said, I consider it problematic that Singular *implicitly* defines the generator degrees when using a matrix order. > If we take the assumption that using weighted monomial orders does not > automatically induces weights (grading or (in your usage) degree) of > variables, then the polynomial ring should not be regarded as a graded ring, > and hence degree() should return the total degree, If we wouldn't view it as a graded ring, degree() would raise a TypeError. I see no reason to change my point of view: * A polynomial ring always is a graded ring, and degree() should act accordingly (that's the case for rings implemented with Singular, but not for the polydict-implementation, which I think is a bug). * The default grading should be given be the exponents of the variables (that's already the case). * It should be possible to define the degrees of the generators independent from the monomial order; that's currently not the case, and I hope I will be able to convince the Singular folks to fix it during the upcoming Sage Days in Kaiserslautern. * A non-default grading should only be used when *explicitly* requested; that's currently not the case. Best regards, Simon -- 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
