2009/3/31 Martin Albrecht <[email protected]>: > > On Tuesday 31 March 2009, Florian wrote: >> Hello everyone, >> >> I've been trying to figure out whether the following functionality is >> implemented, but so far I could not. I was hoping that anyone would >> know if it existed and in that case what the syntax is. >> >> Suppose you computed the reduced Groebner Basis G of an ideal I= >> (f1,...,fn) in some polynomial ring R, and suppose that that Groebner >> Basis turned out to be G=(1). Is there a function that finds some, >> maybe even all, combinations of coefficients h1,...,hn such that >> h1*f1+...+hn*fn=1? >> >> This is basically a byproduct of e.g. the Buchberger Algorithm. The >> question is whether this information can be accessed. > > Like this?
Martin, since this is a frequently asked question, do you think something about this should be added to the groebner_basis docstring? The groebner_basis docstring is 3 pages right now, so this shouldn't be too far down there. Thanks for such extensive documentation for that command already. William > > sage: P.<x,y,z> = PolynomialRing(QQ) > sage: I = Ideal(P.random_element() for _ in range(4)) > sage: I.groebner_basis() > [1] > sage: o = P(1) > sage: o.lift(I.gens()) > ... > > sage: o.lift? > Type: builtin_function_or_method > Base Class: <type 'builtin_function_or_method'> > String Form: <built-in method lift of > sage.rings.polynomial.multi_polynomial_libsingular.MPolynomial_libsingular > object at 0x3003fc8> > Namespace: Interactive > Docstring: > > given an ideal I = (f_1,...,f_r) and some g (== self) in I, > find s_1,...,s_r such that g = s_1 f_1 + ... + s_r f_r > > EXAMPLE: > sage: A.<x,y> = PolynomialRing(QQ,2,order='degrevlex') > sage: I = A.ideal([x^10 + x^9*y^2, y^8 - x^2*y^7 ]) > sage: f = x*y^13 + y^12 > sage: M = f.lift(I) > sage: M > [y^7, x^7*y^2 + x^8 + x^5*y^3 + x^6*y + x^3*y^4 + x^4*y^2 + > x*y^5 + x^2*y^3 + y^4] > sage: sum( map( mul , zip( M, I.gens() ) ) ) == f > True > > Cheers, > Martin > > > > -- > name: Martin Albrecht > _pgp: http://pgp.mit.edu:11371/pks/lookup?op=get&search=0x8EF0DC99 > _otr: 47F43D1A 5D68C36F 468BAEBA 640E8856 D7951CCF > _www: http://www.informatik.uni-bremen.de/~malb > _jab: [email protected] > > > > > -- William Stein Associate Professor of Mathematics University of Washington http://wstein.org --~--~---------~--~----~------------~-------~--~----~ To post to this group, send email to [email protected] To unsubscribe from this group, send email to [email protected] For more options, visit this group at http://groups.google.com/group/sage-support URLs: http://www.sagemath.org -~----------~----~----~----~------~----~------~--~---
