Thanks guys!
On Mar 31, 9:54 am, Martin Albrecht <m...@informatik.uni-bremen.de>
wrote:
> 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?
>
> 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: martinralbre...@jabber.ccc.de
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