Dimensions when intersecting

[wilrop]

I'm having a problem with getting the dimensions right for the diagonal homotopy solver. In the example below, I define two sets of polynomials, where one contains some that are also in the other. Unless I understood it incorrectly, the dimension for the first set is 1 while it is zero for the second. However, when I do this the output is incorrect and gives me results for the variables a1, a2, b1, b2. If I set dim2 = 1, which I think is actually incorrect for this system, the output for the intersection is correct.

polys1 = ['a + b;']
polys2 = ['a + b;', 'a - 5;']
amb_dim = 2
dim1 = 1
dim2 = 0
emb1 = embed(amb_dim, dim1, polys1)
emb2 = embed(amb_dim, dim2, polys2)

sols1 = solve(emb1, verbose=False, precision='d')
sols2 = solve(emb2, verbose=False, precision='d')

emb1[0] = 'a + b - a - b + ' + emb1[0]
emb2[0] = 'a + b - a - b + ' + emb2[0]

polys, sols = diagsolve(amb_dim, dim1, emb1, sols1, dim2, emb2, sols2, prc='d', verbose=False)

print("Diagonal solutions")
for sol in sols:
    print(sol)

polys = ['a + b;', 'a - 5;']
dim3 = 0
emb = embed(amb_dim, dim3, polys)
sols = solve(emb, verbose=False, precision='d')

print("Regular solutions")
for sol in sols:
    print(sol)

Another example is the following,

polys1 = ['x + y + z;']
polys2 = ['z - 2;', 'x + y + z;']
amb_dim = 3
dim1 = 2
dim2 = 1
emb1 = embed(amb_dim, dim1, polys1)
emb2 = embed(amb_dim, dim2, polys2)

sols1 = solve(emb1, verbose=False, precision='d')
sols2 = solve(emb2, verbose=False, precision='d')

emb1[0] = 'x + y + z - x - y - z + ' + emb1[0]
emb2[0] = 'x + y + z - x - y - z + ' + emb2[0]

polys, sols = diagsolve(amb_dim, dim1, emb1, sols1, dim2, emb2, sols2, prc='d', verbose=False)

print("Diagonal solutions")
for sol in sols:
    print(sol)

polys = ['z - 2;', 'x + y + z;']
emb = embed(amb_dim, 1, polys)
sols = solve(emb, verbose=False, precision='d')

print("Regular solutions")
for sol in sols:
    print(sol)

When running this, it gives a solution of dimension 0 for the diagonal homotopy but of dimension 1 for the black box solver. Lastly, the following example gives me zero solutions for the diagonal homotopy but four for the black box solver.

polys1 = [
        '(3 * d * h + 1 * d * j + 9 * d * k + 9 * f * h + 5 * f * j + 3 * f * k + 5 * g * h + 2 * g * j + 1 * g * k) - (5 * d * h + 8 * d * j + 1 * d * k + 5 * f * h + 8 * f * j + 7 * f * k + 2 * g * h + 8 * g * j + 9 * g * k);',
        'b + c - 1;', 'd + f + g - 1;', 'h + j + k - 1;', 'a;']
polys2 = [
    '(3 * a * h + 5 * a * j + 2 * a * k + 6 * b * h + 8 * b * j + 7 * b * k + 4 * c * h + 7 * c * j + 3 * c * k) - (2 * a * h + 1 * a * j + 8 * a * k + 9 * b * h + 7 * b * j + 4 * b * k + 8 * c * h + 2 * c * j + 4 * c * k);',
    'a + b + c - 1;', 'd + g - 1;', 'h + j + k - 1;', 'f;']

amb_dim = 9

dim1 = 4
emb1 = embed(amb_dim, dim1, polys1)
sols1 = solve(emb1, verbose=False, precision='d')

dim2 = 4
emb2 = embed(amb_dim, dim2, polys2)
sols2 = solve(emb2, verbose=False, precision='d')

emb1[0] = 'a + b + c + d + f + g + h + j + k - a - b - c - d - f - g - h - j - k + ' + emb1[0]
emb2[0] = 'a + b + c + d + f + g + h + j + k - a - b - c - d - f - g - h - j - k + ' + emb2[0]

polys, sols = diagsolve(amb_dim, dim1, emb1, sols1, dim2, emb2, sols2, prc='d', verbose=False)

print("Diagonal solutions: ", len(sols))

polys3 = [
    '(3 * d * h + 1 * d * j + 9 * d * k + 9 * f * h + 5 * f * j + 3 * f * k + 5 * g * h + 2 * g * j + 1 * g * k) - (5 * d * h + 8 * d * j + 1 * d * k + 5 * f * h + 8 * f * j + 7 * f * k + 2 * g * h + 8 * g * j + 9 * g * k);',
    'b + c - 1;', 'h + j + k - 1;', 'a;', 'd + g - 1;', 'f;',
    '(3 * a * h + 5 * a * j + 2 * a * k + 6 * b * h + 8 * b * j + 7 * b * k + 4 * c * h + 7 * c * j + 3 * c * k) - (2 * a * h + 1 * a * j + 8 * a * k + 9 * b * h + 7 * b * j + 4 * b * k + 8 * c * h + 2 * c * j + 4 * c * k);',
]

dim = 2
emb = embed(amb_dim, dim, polys3)
sols = solve(emb, verbose=False)

print("Regular solutions: ", len(sols))

All of these examples have in common that some polynomials are overlapping or the same in both systems. I'm wondering then if the output for the diagonal homotopy is a bug or if I should be setting the dimensions differently? I have tried a lot of dimension configurations but anything I try leads to other problems.

I'm honestly completely stuck here, so if you could help me out I would greatly appreciate it!

[janverschelde]

Thanks for reporting this issue. Setting the verbose option to True in the blackbox solver gives different results, I will check this out.
To compute a witness set for one polynomial, there are more efficient methods available.
For example:

from phcpy.sets import witness_set_of_hypersurface as withyp
(p, s) = withyp(3, 'x+y+z;', 'd')

will return in p the embedded system and in s the witness points.