# initialIdeal -- return the initial ideal of a given ideal

## Synopsis

• Usage:
initialIdeal I
• Inputs:
• I, a graded ideal of a polynomial ring
• Outputs:
• an ideal, the initial ideal of the ideal I with default monomial order

## 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

## Ways to use initialIdeal :

• "initialIdeal(Ideal)"

## For the programmer

The object initialIdeal is .