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toricSecantDim -- dimension of a secant of a toric variety

Synopsis

Description

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

See also

Ways to use toricSecantDim :

For the programmer

The object toricSecantDim is a method function.