Description
let $S=K[x_1,\ldots,x_n]$ be a polynomial ring over a field $K$. Let > be a monomial order on $S$. The largest monomial of a polynomial $f\in S$ is called the initial monomial of $f$ and it is denoted by $\mathrm{In}(f).$
If
I is a graded ideal of $S$ then the initial ideal of
I, denoted by $\mathrm{In}(I)$, is the ideal of $S$ generated by the initial terms of elements of
I.
Example:
i1 : S=QQ[x_1..x_5]
o1 = S
o1 : PolynomialRing

i2 : I=ideal {x_1*x_2+x_3*x_4*x_5,x_1*x_3+x_4*x_5,x_2*x_3*x_4}
o2 = ideal (x x x + x x , x x + x x , x x x )
3 4 5 1 2 1 3 4 5 2 3 4
o2 : Ideal of S

i3 : initialIdeal I
2 2 2 2 2
o3 = ideal (x x x , x x x , x x , x x , x x x , x x x , x x )
2 4 5 2 4 5 4 5 1 2 2 3 4 3 4 5 1 3
o3 : Ideal of S
