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TSpreadIdeals > initialIdeal

initialIdeal -- return the initial ideal of a given ideal

Synopsis

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 :

For the programmer

The object initialIdeal is a method function.