an easy Google search give a lot of results about the definition of generic initial ideal. But all definitions I see, are like this one:
I can't use this definition to compute gin(I) even in simple cases, like for ideal $I = (x^2, y^2) ⊂ K[x, y]$. please help me by computing gin($x^2, y^2$), (not using software).
thank you.
see this.
Apply a linear transformation with unknown coefficients (set $x := ax + by, y := cx + dy$) to your ideal. Then find the initial of resulting ideal when you pick coefficients 'at random'. You can do that by calculating its Groebner basis with Buchberger algorithm, just pretend that leading coefficients never become zero (it's true if you pick $a, b, c$ and $d$ at random). Bonus: if you write down the inequalities that you assume, they will give you the set $U$.