toricJoinDim -- Dimension of a join of toric varieties

Synopsis

Usage:

d = toricJoinDim(A,B)

d = toricJoinDim L
Inputs:
- A, a matrix, The A-matrix of a toric variety
- B, a matrix, The A-matrix of a toric variety
- L, a list, A list of A-matrices of toric varieties
Outputs:
- d, an integer, The dimension of the join of the toric varieties defined by the matrices

Description

A randomized algorithm for computing the affine dimension of a join of toric varieties, using Terracini's Lemma.

Each input matrix defines a parameterization of the variety. For each, a vector of parameter values is chosen 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 join 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 : B = matrix{{1,1,1,1,1}}

o2 = | 1 1 1 1 1 |

              1        5
o2 : Matrix ZZ  <--- ZZ

i3 : toricJoinDim(A,B)

o3 = 3

i4 : toricJoinDim(B,B)

o4 = 1

Caveat

All input matrices must have the same number of columns.

Ways to use `toricJoinDim` :

"toricJoinDim(List)"
"toricJoinDim(Matrix,Matrix)"

For the programmer

The object toricJoinDim is a method function.