On 25 Mai, 08:18, Rob Beezer <goo...@beezer.cotse.net> wrote: > I'm wondering if Sage has the following function for polynomials over > a field. Mostly, I don't know if it is a common thing to expect, and > if so, I have no idea what it would be called. So a pointer would be > of real help, if it exists. The paper I am cribbing from is about > matrices over ZZ, but I suspect it generalizes to any PID, so I may > not be translating everything exactly right (eg the degree condition). > > Input: p, q, r in F[x] with gcd(p, q, r) = 1 > > Output: s in F[x] such that gcd(p + sq, r) = 1, degree(s) < > degree(r) > > Thanks, > Rob
Hi Rob, a partial (mathematicians) answer: the property in question is the defining property of "Euclidean rings" (look e.g. in van der Waerden's "Algebra I"). By definiton, such a ring is (or can be) equipped with a "degree" function such that the usual "Euclidean algorithm" can be made to work. Euclidean rings can easily shown to be PIDs (but there are PIDs that can't be equipped with any such a "Euclidean degree" function). Prime examples are the rational integers (degree == absolute value) and univariate polynomial rings over fields (degree == degree as polynomial). Now, what I do not know is whether in the hierarchy of rings in Sage, Euclidean rings show up as ancestors of PIDs, or whether there is some property "IsEuclidean()" and if so, "EuclideanDegreeFunction()", "EuclideanGCD()", "EuclideanXGCD()" or such. Hope this helps anyway! Cheers, Georg -- 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
