On Mon, Feb 21, 2011 at 1:40 PM, Nils Bruin <nbr...@sfu.ca> wrote: > On Feb 21, 10:31 am, Robert Bradshaw <rober...@math.washington.edu> > wrote: >> k[x,y], k[x,z], k[y,z] all coerce to k[x,y,z], but there some overhead >> due to the extra object creation (assuming one needed to coerce back >> into the larger ring). This, and creation of the new ring, is probably >> still pretty cheap. > > The problem I see is that the variable names are not the only > parameters involved in creating a new polynomial ring. How about term > ordering, underlying implementation etc? At some point I bet there are > going to be polynomial ring k[x,y,z] with custom term orderings.
Sage has had multivariate polynomial rings with a wide range of term orderings since 2005. > If > you compute a resultant of two elements there with respect to z, what > kind of k[x,y] are you going to create to hold the answer? The choice of term ordering on k[x,y,z] induces one on k[x,y], since the set of monomials in k[x,y] is a subset of that in k[x,y,z]. If one types f.resultant(g,...) then this shouldn't work unless f and g canonically coerce to a common parent (or perhaps, g gets coerced into f.parent()). > This problem does not arise in the univariate case, because the > prospective parent of the resultant already exists (i.e., is the base > ring). > > Therefore, I think the safer choice is to let resultants of > multivariate polynomials end up in the same parent again. If people > want it in another ring, they can create that before hand and coerce > it in (apparently cheaply). I'm not sure if letting the resulting be in the same parent is safe or not. However, two points in favor of the choice you suggest are: (1) It is the current choice in Sage, and leaving it that way is very good from the point of view of backward compatibility. I like maintaining backward compatibility. (2) It is also the choice that is made in Magma, so leaving it as is now is better for compatibility with Magma: > R<x,y,z> := PolynomialRing(RationalField(),3); > Parent(Resultant(x,y,2)); Polynomial ring of rank 3 over Rational Field Order: Lexicographical Variables: x, y, z --- What we have now: ---------------------------------------------------------------------- | Sage Version 4.6.1, Release Date: 2011-01-11 | ---------------------------------------------------------------------- sage: R.<x,y,z> = QQ[] sage: parent(x.resultant(y,z)) Multivariate Polynomial Ring in x, y, z over Rational Field -- William Stein Professor of Mathematics University of Washington http://wstein.org -- 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
