How to find the generic initial ideal?

Question

Here is an example from Ezra Miller's book: Combinatorial Commutative Algebra,p26-27

Let $f,g\in k[x_1,x_2,x_3,x_4]$ be a 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 ideals $J=\operatorname{gin}_{\operatorname{lex}}(I)$ are $(x_2^4,x_1x_3^2,x_1x_2,x_1^2)$, the ideal $J=\operatorname{gin}_{\operatorname{revlex}}(I)$ are $(x_2^3,x_1x_2,x_1^2)$.

How to find it?

Answer 1

You should apply a generic linear coordinate transform to the ideal and then compute the initial ideal. The matrices for which the result is the generic initial ideal is a (Zariski) open subset (Lemma 2.6, in the book). In particular the complement is of lower dimension.
Leaving distribution issues aside, if you 'pick a random coordinate transform', then you'll always find the generic initial ideal. In practice you'll find it with high probability. There is also an implementation in Macaulay2

Edit: Here is one way to computer-prove the statements from the book. To do this I make a ring in which the coefficients of $f,g$ are extra variables. There are 10 degree 2 monomials in four variables, therefore we need a ring with 10+10+4 variables. Here is the Macaulay2 code:

S = QQ[a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,b1,b2,b3,b4,b5,b6,b7,b8,b9,b10, x1,x2,x3,x4]
ba = {x1^2, x1*x2, x1*x3, x1*x4, x2^2, x2*x3, x2*x4, x3^2, x3*x4, x4^2}
a={a1,a2,a3,a4,a5,a6,a7,a8,a9,a10}  -- coefficients of f
b={b1,b2,b3,b4,b5,b6,b7,b8,b9,b10}  -- coefficients of g
f = sum (for i to 9 list a#i*ba#i)
g = sum (for i to 9 list b#i*ba#i)
I = ideal (f,g)
monomialIdeal I  -- returns the initial ideal

Running it computes the revlex initial ideal because revlex is the default order. The output is $(a_1 x_1^2,b_1 x_1^2,a_2 b_1x_1 x_2,a_5 b_1 x_1 x_2^2,a_1 a_5 b_2 x_1 x_2^2,a_5^2 b_1^2 x_2^3)$ and thus making the $a_i$ and $b_i$ invertible we find the claimed result. For lex you have to change the variable and monomial order on $S$. One has to be a bit careful with the ordering of the variables, because M2 does not know that the $a_i$ and $b_i$ are invertible. Putting the $x_1$ last does the right thing for revlex(this needs a quick check). The Macaulay2 package that I linked is for a concrete ideal (not the generic complete intersection) like in the example.

Answer 2

The general problem of computing invariants of the lexicographic generic initial ideal of an ideal is discussed in the last chapter of Mark Green Barcelona's notes

There are papers devoted in particular to the study of gin-lex of ideals defining curves (not only complete intersections) see for example math/0402418 and arXiv:1108.0045.