Let $F, f_1, \ldots f_5$ be polynomials in $\mathbb{Z}_p[r,s,t,u,v]$, the ring of polynomials in 5 variables over the integers modulo an odd prime $p$.
By forming the ideal $J:=< f_1, \ldots f_5>$ I can test whether $F$ is a member of $J$. Indeed $F$ is a member of $J$ and so I know there exists polynomials $a_1,\dots,a_r \in \mathbb{Z}_p[r,s,t,u,v]$ such that $$F = a_1f_1+\dots+ a_rf_r
$$
My question is how to explicitly compute $a_1,\dots,a_r$ in Maple, or Sage if you prefer. Thank you very much for any help you can give.
