Hi! On 21 Sep., 11:05, Kwankyu Lee <ekwan...@gmail.com> wrote: > sage: tord = TermOrder(matrix([3,2,4,1,1,0,1,0,0])) > sage: S.<t>=PolynomialRing(QQ) > sage: R.<x,y,z> = PolynomialRing(S,'x',3,order=tord) > sage: (x^2).degree() > 2
I think that's a bug. > So the behavior is not consistent among different backend engines, namely, > Singular and PolyDict. I think degree() should return the total degree, not > the weighted degree... I never take it for granted that the generators of a graded ring (e.g., a polynomial ring) are in degree 1. So, perhaps my point of view is a bit odd. IMHO, there only is one notion, namely "degree". I would actually still call it a "degree" if it is not a natural number. Say, you have a ZZ/2-graded ring. Compare with quantum groups, where you may have dimensions that are complex numbers, not natural numbers. Neither the notions of degree nor of dimension are reserved for NN. There are two situations where I might add the word "weighted" to notions like "degree" or "homogeneous": (1) If it is a generalisation from "generators are degree one" to "generators can have arbitrary degree". Example: #7797, where in the beginning everything had to be homogeneous with generators in degree one, and I generalised it so that one still has to work with homogeneous ideals, but the generators of the ring are in any non- negative integral degree. So, I used "weighted homogeneous" just to emphasize the difference. But really, I think the word "homogeneous" should be clear enough. (2) If the degrees temporarily change. For example, if R is QQ[x,y] with generators in degree 2 and 3, and p=x*y, and I temporarily change the order so that x and y are of degree 1, then I would say: p is of degree 5 (namely as an element of R) and of weighted degree 2. > I am not sure whether it is always desirable that the weighted > term order determines the weights of the variables. I think in principle > they can be independent. The degrees of the generators belong to the definition of a graded ring. Thus, if one wants to have a ring with generators in degree 1 then one should do so. However, I do agree that choosing a matrix order should not *implicitly* change the degrees of the ring generators. 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
