`toric_ideal(pts::ZZMatrix)` is (often) very inefficient

[imkhln]

Computing the toric ideal of a tuple of lattice points directly as a homomorphism kernel is much faster than applying toric_ideal to the respective integer matrix. In virtually all cases that are not too small. A random example with 7 points in dimension 2:

A = [[9, 1], [3, 3], [9, 8], [5, 1], [7, 10], [5, 5], [1, 5]]
R, x = polynomial_ring(QQ, "x" => (1:7))
S, z = polynomial_ring(QQ, "z" => (1:2))
f = hom(R, S, [prod(z.^A[i]) for i in (1:7)])
I = kernel(f)

runs almost instantly but a subsequent call of toric_ideal(matrix(ZZ, A)) does not terminate after several minutes. And this happens for all randomly generated examples with the same or larger parameters. (AFAIK, these two calls should output the same ideal, and in small cases where both work fast this is indeed the case.)

This was tested in Oscar 1.3.0 and also in some of the older versions, probably not a recently introduced issue.

[YueRen]

I think the problem is caused by the following lines:

Oscar.jl/src/AlgebraicGeometry/ToricVarieties/NormalToricVarieties/toric_ideal.jl

Lines 108 to 110 in c0fcef9

	function toric_ideal(R::MPolyRing, pts::ZZMatrix)
	intkernel = kernel(pts, side=:left)
	presat = binomial_exponents_to_ideal(R, intkernel)

Because the kernel is computed over ZZ, the result intkernel has enormous entries, and the resulting ideal presat has enormous exponents:

julia> intkernel = kernel(matrix(QQ,A), side=:left)
[-1//8    -21//8   1   0   0   0   0]
[-1//2     -1//6   0   1   0   0   0]
[ 3//8   -83//24   0   0   1   0   0]
[    0     -5//3   0   0   0   1   0]
[ 1//2    -11//6   0   0   0   0   1]

julia> intkernel = kernel(matrix(ZZ,A), side=:left)
[  -1     -21     8     0   0   0   0]
[  -3     -32    12     3   0   0   0]
[-176   -1992   747   166   1   0   0]
[ -85    -960   360    80   0   1   0]
[ -93   -1056   396    88   0   0   1]

julia> presat = binomial_exponents_to_ideal(R, intkernel)
Ideal generated by
  -x[1]*x[2]^21 + x[3]^8
  -x[1]^3*x[2]^32 + x[3]^12*x[4]^3
  -x[1]^176*x[2]^1992 + x[3]^747*x[4]^166*x[5]
  -x[1]^85*x[2]^960 + x[3]^360*x[4]^80*x[6]
  -x[1]^93*x[2]^1056 + x[3]^396*x[4]^88*x[7]

@HereAround Is a the kernel over ZZ necessary or would an integer matrix from the kernel over QQ suffice?

[HereAround]

Computing the toric ideal of a tuple of lattice points directly as a homomorphism kernel is much faster than applying toric_ideal to the respective integer matrix. In virtually all cases that are not too small. A random example with 7 points in dimension 2:

A = [[9, 1], [3, 3], [9, 8], [5, 1], [7, 10], [5, 5], [1, 5]]
R, x = polynomial_ring(QQ, "x" => (1:7))
S, z = polynomial_ring(QQ, "z" => (1:2))
f = hom(R, S, [prod(z.^A[i]) for i in (1:7)])
I = kernel(f)

runs almost instantly but a subsequent call of toric_ideal(matrix(ZZ, A)) does not terminate after several minutes. And this happens for all randomly generated examples with the same or larger parameters. (AFAIK, these two calls should output the same ideal, and in small cases where both work fast this is indeed the case.)

This was tested in Oscar 1.3.0 and also in some of the older versions, probably not a recently introduced issue.

Thank you @imkhln for pointing this out to us. (Also, apologies for my slow response.)

@lkastner will work on improving the toric ideal computation along the lines you suggested.