A randomized algorithm for computing the affine dimension of a secant of a toric variety using Terracini's Lemma.
Here the kth secant means the join of k copies of I. Setting k to 1 gives the dimension of the ideal, while 2 is the usual secant, and higher values correspond to higher order secants.
The matrix A defines a parameterization of the variety. The algorithm chooses k vectors of parameter values at random from a large finite field. The dimension of the sum of the tangent spaces at those points is computed.
This algorithm is much much faster than computing the secant variety.
i1 : A = matrix{{4,3,2,1,0},{0,1,2,3,4}}
o1 = | 4 3 2 1 0 |
| 0 1 2 3 4 |
2 5
o1 : Matrix ZZ <--- ZZ
|
i2 : toricSecantDim(A,1) o2 = 2 |
i3 : toricSecantDim(A,2) o3 = 4 |
i4 : toricSecantDim(A,3) o4 = 5 |
i5 : toricSecantDim(A,4) o5 = 5 |
The object toricSecantDim is a method function.