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Find specific linear combination in multivariate polynomial ring

asked 2012-03-25 01:53:55 -0500

Martin Brandenburg gravatar image

updated 2012-03-25 02:56:39 -0500

DSM gravatar image

Assume that I have given a sequence of polynomials $f_1,\dotsc,f_s$ in a multivariate polynomial ring (over $\mathbb{Z}$, if that matters) and want to decide whether a given polynomial $g$ can be written as $g = \lambda_1 f_1 + \dotsc + \lambda_s f_s$. Then in Sage I just let

I = Ideal([f_1,...,f_s])

and test with

g in I

If this returns True, how can I get Sage to display some possible $\lambda_1,\dotsc,\lambda_s$?

As for my specific problem, I have already tried it by hand, but this is hard: My polynomial ring has $15$ indeterminates and there are $s = 250$ polynomials.

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There is a related question http://ask.sagemath.org/question/1064/explicit-representation-of-element-of-ideal which answers my question if the base ring was a field.

Martin Brandenburg gravatar imageMartin Brandenburg ( 2012-03-25 01:59:30 -0500 )edit

I could solve my problem by feeding sage with base fields such as $\mathbb{Q}$ and $\mathbb{F}_2$ and experimental comparing of the results, to get a correct linear combination over the base ring $\mathbb{Z}$. But I think it is interesting whether there is a general method implemented.

Martin Brandenburg gravatar imageMartin Brandenburg ( 2012-03-25 05:01:23 -0500 )edit

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answered 2012-03-25 11:28:22 -0500

AFAIK, Singular can handle this case but the Sage wrappers restrict the coefficient domain to a field. You can work around this with the magical Singular function interface. Using the example from the previous question linked above:

sage: R.<x,y,z,t> = ZZ[]
sage: C1= 17*x^2 + 7*y^2 - 26*y*z + 7*z^2
sage: C2= 13*y^2 - 7*y*z + 13*z^2 - 51*t^2
sage: I = (C1, C2)*R
sage: f in I
False
sage: 221*f in I
True
sage: lift = sage.libs.singular.ff.lift
sage: lift(I, 221*f)
[         13*x^2 + 17*y*z - 21*t^2]
[-7*x^2 + 17*y^2 + 17*z^2 + 78*t^2]
sage: (13*x^2 + 17*y*z - 21*t^2)*C1 + (-7*x^2 + 17*y^2 + 17*z^2 + 78*t^2)*C2
221*x^4 + 221*y^4 + 221*z^4 - 3978*t^4
sage: f
x^4 + y^4 + z^4 - 18*t^4
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Thanks!

Martin Brandenburg gravatar imageMartin Brandenburg ( 2012-03-27 02:52:37 -0500 )edit

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Asked: 2012-03-25 01:53:55 -0500

Seen: 531 times

Last updated: Mar 25 '12