Let $R=k[[x,y,z]]$ and $I=(xz-y^3,yz-x^4,z^2-x^3y^2)$ be an ideal. I am trying to compute the ideal $I^*$ of initial forms of $I$ in the associated graded ring $\mathrm{gr}_m(R)=\bigoplus_{n\geq 0} m ^n/m^{n+1}$ where $m$ is the irrelevant ideal of $R$.
First, we know that $xz=\mathrm{in}(xz-y^3)$, $yz=\mathrm{in}(yz-x^4)$, $z^2=\mathrm{in}(z^2-x^3y^2) \in I^*$. Since all of them are multiple of $z$, they can not generate $I^*$. By writing an element in $I$ as $f(xz-y^3)+g(yz-x^4)+h(z^2-x^3y^2)$, we are able to find that when $f=y$, $g=x$, we have a new generator for $I^*$, which is $y^4$, so that $I^*=(xz,yz,z^2,y^4)$. (Is it correct?)
My question is: do we have a general way (or algorithm) to find ideal of initial forms (let's first consider the case when we have ideals in formal power ring over a field) or do we need to do case by case? Any hint appreciated. Thank you!