I'm looking for the algorithms associated with the two Magma functions PolynomialWithFixedBasis and Coordinates. In particular, if $g_1,\ldots,g_n$ is a Gröbner Basis (in decreasing order), I would like to compute some polynomial coordinates $\{h_{i,j}\}$ such that whenever $g_i \in \langle g_{i+1},\ldots,g_n\rangle$, we have $g_i=\sum_{j=i+1}^n{h_{i,j}g_j}$. The two functions PolynomialWithFixedBasis and Coordinates do the job, with
$$I:=\text{Ideal With Fixed Basis}([G[j] : j\ in\ [i+1..n]]);$$
$$H:=\text{Coordinates}(I,G[i]);$$
But I would like to understand how they work. Thank you!