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FourTiTwo :: toricMarkov

toricMarkov -- calculates a generating set of the toric ideal I_A, given A; invokes "markov" from 4ti2

Synopsis

Description

Suppose we would like to comput the toric ideal defining the variety parametrized by the following matrix:

A = matrix"1,1,1,1;0,1,2,3"

Since there are 4 columns, the ideal will live in the polynomial ring with 4 variables.

R = QQ[a..d]
M = toricMarkov(A)

Note that rows of M are the exponents of minimal generators of IA. To get the ideal, we can do the following:

I = toBinomial(M,R)

Alternately, we might wish to give a lattice basis ideal instead of the matrix A. The lattice basis will be specified by a matrix, as follows:

B = syz A
N = toricMarkov(transpose B, InputType => "lattice")
J = toBinomial(N,R) -- toricMarkov(transpose B, R, InputType => "lattice")

We can see that the two ideals are equal:

I == J

Also, notice that instead of the sequence of commands above, we could have used the following:

toricMarkov(A,R)

Ways to use toricMarkov :