Let $k$ be a field, $m$ and $n$ be positive integers, and suppose $f$ and $g$ are two polynomials in $k[x_1,\ldots,x_n]$ such that $f^{i-1}$ divides $g^{i}$ for all $i=2,\ldots,m$. Consider the subalgebra $A$ in $k[x_1,\ldots,x_n]$ generated by the elements $f$, $g$, and $g^if^{1-i}$ for $i=2,\ldots,m$.
I'd like to write down all of the relations between the generators of $A$, that is, if we set $\lambda_i:=g^if^{1-i}$ for $i=0,\ldots,m$, then we have
$$A\cong k[\lambda_i]/I$$
for some ideal $I$. What are the generators of $I$? Through preliminary study, I've conjectured that $I$ is generated by all polynomials of the form $\lambda_i\lambda_{k-i}-\lambda_{i+1}\lambda_{k-i-1}$ for all $0\le i<k\le m$, but I can't seem to prove that these polynomials generate $I$.
Have I written down a generating set? Has anyone seen this variety before?