# toricSecantDim -- dimension of a secant of a toric variety

## Synopsis

• Usage:
toricSecantDim(A,k)
• Inputs:
• A, , the A-matrix of a toric variety
• k, an integer, order of the secant
• Outputs:
• an integer, the dimension of the kth secant of variety defined by matrix A

## 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