toricIdeal -- returns a toric ideal

Synopsis

Usage:

toricIdeal(A,R)
Inputs:
- A, a matrix, a d x n integer matrix
- R, a ring, a polynomial ring in n variables
Outputs:
- an ideal, the toric ideal associated to A in R

Description

Given a $d\times n$ Matrix A and a polynomial ring in $n$ variables $R$, this method returns the toric ideal associated to $A$ in $R$. To do this, toricIdeal saturates the lattice basis ideal of the kernel of $A$ with respect to the product of the variables of $R$.

i1 : A=matrix{{1,1,1,1,1,1},{1,2,1,2,3,0},{0,2,2,0,1,1}}

o1 = | 1 1 1 1 1 1 |
     | 1 2 1 2 3 0 |
     | 0 2 2 0 1 1 |

              3       6
o1 : Matrix ZZ  <-- ZZ

i2 : R=QQ[a..f]

o2 = R

o2 : PolynomialRing

i3 : toricIdeal(A,R)

                                   3     3      2     2      2         2  
o3 = ideal (c*d - e*f, a*b - e*f, c e - b f, a*c e - b d*f, a c*e - b*d f,
     ------------------------------------------------------------------------
      3     3      3    2 2   2 2        2   3     2 2     3    2 2   2 2  
     a e - d f, b*d  - a e , b d  - a*c*e , b d - c e , a*c  - b f , a c  -
     ------------------------------------------------------------------------
          2   3     2 2
     b*d*f , a c - d f )

o3 : Ideal of R

i4 : A=matrix{{1,1,1,1,1},{0,0,1,1,0},{0,1,1,0,-2}}

o4 = | 1 1 1 1 1  |
     | 0 0 1 1 0  |
     | 0 1 1 0 -2 |

              3       5
o4 : Matrix ZZ  <-- ZZ

i5 : R=toGradedRing(A,QQ[a..e])

o5 = R

o5 : PolynomialRing

i6 : toricIdeal(A,R)

                          2    2    2            3    2
o6 = ideal (a*c - b*d, a*d  - c e, a d - b*c*e, a  - b e)

o6 : Ideal of R

Ways to use toricIdeal :

toricIdeal(Matrix,Ring)

For the programmer

The object toricIdeal is a method function.