findANonZeroMinor -- finds a non-vanishing minor at some randomly chosen point

Synopsis

• Usage:

  findANonZeroMinor(n,M,I)

• Inputs:
  • I, an ideal, in a polynomial ring over QQ or ZZ/p for p prime
  • M, a matrix, over the polynomial ring
  • n, an integer, the size of the minors to consider
• Optional inputs:
  • Strategy => a symbol, default value Default, to specify which strategy to use when calling randomPoints
  • Verbose => a Boolean value, default value false, set to true for verbose output
  • Homogeneous => a Boolean value, default value false, controls if the computations are homogeneous (in calls to randomPoints)
  • MinorPointAttempts => an integer, default value 5, how many points to check the rank of the matrix at
  • DimensionFunction => a function, default value dim, specify a custom dimension function, such as the default dimViaBezout or the Macaulay2 function dim
  • DecompositionStrategy => ..., default value null, control how ideals of points are factored into minimal primes
  • ExtendField => ..., default value true, an option used to specify if extending the finite field is permissible here
  • NumThreadsToUse => ..., default value 1, number of threads that the function will use in a brute force search for a point
  • PointCheckAttempts => ..., default value 0, Number of times that the function will search for a point
  • Replacement => ..., default value Binomial, When changing coordinates, whether to replace variables by general degree 1 forms, binomials, etc.
• Outputs:
  • a sequence, The functions outputs the following: 1. randomly chosen point $P$ in $V(I)$, 2. the indexes of the columns of $M$ that stay linearly independent upon plugging $P$ into $M$, 3. the indices of the linearly independent rows of the matrix extracted from $M$ using (2), 4. a random $n\times n$ submatrix of $M$ that has full rank at $P$.

Description

i1 : R = ZZ/5[x,y,z];

i2 : I = ideal(random(3,R)-2, random(2,R));

o2 : Ideal of R

i3 : M = jacobian(I);

             3      2
o3 : Matrix R  <-- R

i4 : findANonZeroMinor(2,M,I)

o4 = ({-2, 0, 2}, {0, 1}, {0, 1}, {1} | -2z -x2-2y2+2xz+2z2 |)
                                  {1} | 2z  xy-y2-2yz-2z2   |

o4 : Sequence

See also

• randomPoints -- a function to find random points in a variety.