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

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!