maxMinors -- computes the ideal generated by the maximal non-vanishing minors of a given matrix

Synopsis

• Usage:

  maxMinors M

• Inputs:
  • M, a matrix, with entries in a ring R.
• Outputs:
  • an ideal, the ideal generated by maximal non-vanishing minors of the matrix M.

Description

Let I_t(M) be the ideal in R generated by the t × t minors of M. If there exists an r such that I_r(M) is non-zero and I_r+1(φ) = 0, then maxMinors M gives I_r(M).

i1 : R = QQ[x,y]

o1 = R

o1 : PolynomialRing

i2 : M = matrix{{x,0},{-y,x},{0,-y}}

o2 = | x  0  |
     | -y x  |
     | 0  -y |

             3       2
o2 : Matrix R  <--- R

i3 : maxMinors M

             2         2
o3 = ideal (x , -x*y, y )

o3 : Ideal of R

This method returns the unit ideal as the ideal of maximal minors of the zero matrix.

i4 : N = matrix{{0_R}}

o4 = 0

             1       1
o4 : Matrix R  <--- R

i5 : maxMinors N

o5 = ideal 1

o5 : Ideal of R

See also

• minors -- ideal generated by minors