# Find specific linear combination in multivariate polynomial ring

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.

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.

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.