William mentioned an area dear to my heart: On Jun 5, 10:47 pm, William Stein <wst...@gmail.com> wrote: > * Magma's linear algebra over p-adics is better than in Sage. > > PROJECT: Design and implement good algorithms for linear algebra over > p-adics (subtle and interesting research area). > There is actually some room for improvement in Magma's handling of p- adics. Consider the following example:
> F := Qp(3, 10); > M := Matrix([[F!1, F!2, F!1], [F!1, F!5, F!4], [F!4, F!8, F!1]]); > CharacteristicPolynomial(M); $.1^3 - (7 + O(3^10))*$.1^2 - (3^3 + O(3^10))*$.1 + 3^2 + O(3^10) This is valid but not best possible. If you do a bit of row reduction on the matrix you get [1 + O(3^10) 2 + O(3^10) 1 + O(3^10)] [0 + O(3^10) 3 + O(3^10) 3 + O(3^10)] [3 + O(3^10) 6 + O(3^10) 0 + O(3^10)] from which it is not hard to see that the determinant of M is actually 3^2 + O(3^11). (Divide the second and third rows by 3, and then you can do all the arithmetic in Z/3^9 Z.) The best way to handle this seems to be using Newton polygons, which will have to wait for roed's p-adic polynomials code to come online. After that, there is a nice project waiting here, which has important applications to computing zeta functions of varieties using p-adic cohomology (in which it is crucial to use as little working precision as you can get away with, for efficiency). Kiran --~--~---------~--~----~------------~-------~--~----~ To post to this group, send email to sage-devel@googlegroups.com To unsubscribe from this group, send email to sage-devel-unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/sage-devel URLs: http://www.sagemath.org -~----------~----~----~----~------~----~------~--~---
