How to find polynomial multipliers that relate a specific element in the Groebner basis to the original generators?

Question

If I have a particular set of generators for an ideal, (eg. $f_1$, $f_2$, $\ldots$, $f_n$), it is obviously straightforward to compute the Groebner basis for this ideal (say, $g_1$, $g_2$, $\ldots$, $g_m$) using an algebra package such as Singular/SAGE/Maple, etc. What I want to know is, is there a way to use a symbolic algebra package (eg. Singular, SAGE, etc) to compute the polynomial multipliers that connect a specific polynomial in the Groebner basis, say $g_1$, to the original generators? In other words, if I've already computed $g_1$, $g_2$, $\ldots$, $g_m$, I'd like to find $h_1$, $h_2$, $\ldots$ $h_n$ such that $g_1 = h_1f_1 + h_2f_2 + \ldots + h_nf_n$.

I haven't yet been able to find a way to compute these polynomials ($h_i$) automatically using any algebra package I'm currently aware of. Is the only way forward to do the Buchberger algorithm manually?

Thank you so much in advance for any help/insight anyone might be able to offer on this problem!

Answer 1

In Maple, if you use the Groebner:-Basis command with the option output=extended, you get in addition to the Groebner basis a list of lists giving coefficients of each basis element for the generators.

Answer 2

Update: I've found a solution!! The command to use in both Singular and SAGE is 'lift'. I thought I'd post this here, just in case this helps anyone else :)

Eg, in Singular, if we have three generators for an ideal - f1, f2, f3 (say) - we can generate the ideal with the command 'ideal I = f1, f2, f3;' Then, for ANY polynomial p in that ideal (including but not limited to an element in the Groebner Basis), we can get the multipliers h1, h2, f3 (where p = h1*f1 + h2*f2 + h3*f3), using the command 'lift(I,p)'.

SAGE has a similar command.