Obtaining explicit homotopies from a PolyhedralTracker

[oskarhenriksson]

Is there a way to get explicit formulas for the polyhedral homotopies (or the underlying mixed subdivision) associated with a PolyhedralTracker and PolyhedralStartSolutionsIterator obtained from a square system obtained in the following way?

tracker, start_solution = polyhedral(F)

[PBrdng]

Yes, this is possible. A PolyhedralTracker contains support and lifting that are used to compute the mixed cells using MixedSubdivision.jl. You can read the documentation for computing mixed cells directly.

The PolyhedralStartSolutionsIterator, in fact, computes mixed cells iteratively. It returns the start solutions without the cells. Nevertheless, looking at the code you can also see how to use mixed cells.

[oskarhenriksson]

Thanks, that sounds like what I want! However, I seem to be doing something wrong when I try to recover the lifting, since I just get zero vectors. Do you have an idea of what's going on?

julia> @var x[1:3];
julia> F = System([x[1]^2+x[2]+x[1]*x[2]+1,x[2]^3-x[1]*x]3]+2,x[1]*x[2]*x[3]+1]);
julia> tracker, starts = polyhedral(F, only_non_zero=true);
julia> display(tracker.lifting)

3-element Vector{Vector{Int32}}:
 [0, 0, 0, 0]
 [0, 0, 0]
 [0]

[oskarhenriksson]

I think I understand it now! The choice of lifting is made by compute_mixed_cells when the first instance of the starts iterator is generated. This updates starts.lifting, which through a shared reference also leads to tracker.lifting getting updated.

Example:

@var x[1:3]
F = System([x[1]^2+x[2]+x[1]*x[2]+1,x[2]^3-x[1]*x]3]+2,x[1]*x[2]*x[3]+1])
tracker, starts = polyhedral(F, only_non_zero=true)
display(tracker.lifting)
first(starts)
display(tracker.lifting)

3-element Vector{Vector{Int32}}:
 [0, 0, 0, 0]
 [0, 0, 0]
 [0, 0]

3-element Vector{Vector{Int32}}:
 [-66, -632, -1723, -458]
 [-1243, 1129, -441]
 [-98, 1671]

[saschatimme]

You can also simply disregard the returned start solution iterator from polyehdral and provide your own instance of PolyhedralStartSolutionsIterator where you provide the lifting and associated fine mixed cells you are interested in (the MixedSubdivisions package can compute these). Just make sure that you are using a consistent order of the columns in the support matrices.

[oskarhenriksson]

Thanks @saschatimme!