makeGradedRing -- makes a polynomial ring graded by a matrix

Synopsis

Usage:

makeGradedRing A
Inputs:
- A, a d x n integer matrix
- x, the variable of the multigraded polynomial ring
Outputs:
- a polynomial ring, a multigraded polynomial ring in n variables

Description

This method takes a $d\times n$ integer matrix $A$ and makes the polynomial ring $\QQ[x_0,..,x_{n-1}]$ with the degree of the i-th variable being the i-th column of $A$.

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

o1 = | 1 1 1 1 1  |
     | 0 0 1 1 0  |
     | 0 1 1 0 -2 |

              3       5
o1 : Matrix ZZ  <-- ZZ

i2 : R = makeGradedRing(A,t)

o2 = R

o2 : PolynomialRing

We can see that $R$ is graded by the columns of $A$

i3 : describe R

o3 = QQ[t ..t , Degrees => {{1}, {1}, {1}, {1}, {1 }}, Heft => {1, 2:0}]
         0   4              {0}  {0}  {1}  {1}  {0 }
                            {0}  {1}  {1}  {0}  {-2}

Ways to use makeGradedRing :

makeGradedRing(Matrix,Symbol)

For the programmer

The object makeGradedRing is a method function.