I have come across results/algorithms on how, given a set of polynomials in $k[x_1,\ldots,x_n]$ ($k$ a field) to find the ideal of relations. E.g. $k[a^2,ab,c^2] \cong k[x,y,z]/(xy-z^2)$
I am interested in going backwards, given an ideal (of relations) $I$ and $A=k[x_1,\ldots,x_n]/I$, is there an nice/good way of finding polynomials $P_i(x_1\ldots,x_n)$ in $A$ so that $I$ is their ideal of relations is $I$. e.g. in $k[x_1,\ldots,x_5]/(x_1x_2x_3-x_3x_4^{2}-x_5^2) \cong k[P_1,\ldots,P_5]$
Of course I suppose you can just write down $P_i(x_j) = \sum_n c_{i,n} \prod x_j^{e_j}$ and equate them and try to solve for coefficients and see if there is a solution, but that seems very inefficient, and really a pain to do by hand for my example in $5$ variables when the ideal is not monomial.
Another way of asking: (I just found out while writing this question): one can use Grobner bases to find the ideal defining the relations between given polynomials. The questions is can that process be reversed?