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Here is an example from Ezra Miller's book: Combinatorial Commutative Algebra, pp. 26-27.

Let $f,g\in k[x_1,x_2,x_3,x_4]$ be generic forms of degree $d$ and $e$. The generic initial ideal of $I=\langle f,g\rangle$ for both the lexicographic order and the inverse lexicographic order when $(d,e)=(2,2)$: the ideal $J=\operatorname{gin}_{\operatorname{lex}}(I)$ is $(x_2^4,x_1x_3^2,x_1x_2,x_1^2)$, the ideal $J=\operatorname{gin}_{\operatorname{revlex}}(I)$ is $(x_2^3,x_1x_2,x_1^2)$.

How to find it?

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How is $\operatorname{gin}_{\operatorname{lex}}$ defined? – Davide Giraudo Apr 8 '12 at 8:51
Rough speaking,lex is a kind of order <,the initial ideal in$_<$(gI) as a function of g is a constant on a Zariski open subset U in GL$_n$ is called a gin for I and <.I think it is a mainly part of the initial ideal,but I just learn it some days ago,so cannot say anything deeply. – Strongart Apr 9 '12 at 10:44

You can use a software such as Macaulay2. here are some examples.

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