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: [email protected]
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