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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!

user26857
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mathmathmath
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    In your case, you may assume $R = k[x,y,z]$ as both $R$ and its completion at the maximal ideal $(x,y,z)$ have the same tangent cone. An algorithmic way to compute $I^*$ is using Grobner basis. If you just want to check your calculation, you can use computer algebra software such as Macaulay2, Singular, and etc. With Macaulay2, the command "tangentCone" does the job. – Youngsu Aug 22 '19 at 15:11
  • Hi Youngsu, thank you for your hint. I was thinking if you meant computing the initial ideal (with respect to a monomial order) by "using Groebner basis"? If so, is there any monomial order that give us the least total degree monomial as the initial form? – mathmathmath Aug 26 '19 at 20:50
  • It is not a monomial order. You need to use a term order. – Youngsu Aug 26 '19 at 22:43

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