Spine

spine(f::MP.AbstractPolynomial; options...)

Compute the spine of the amoeba $\mathcal{A}(f)$. This algorithm computes first an approximation of $\mathcal{A}(f)$ and from this the spine. Returns a Spine2D.

Example

@polyvar x y
# use all the defaults
spine(x^2 + y^2 + 1)

# Maybe we think that the compleement of the amoeba has some very small components
# and we want to use an explicit domain
spine(x^2 + y^2 + 1, domain=(-5, 5, -5, 5), minimal_component_size=0.001)

Optional arguments:

• minimal_component_size=0.01: A guarantee that in each component of the complement of $\mathcal{A}(f)$ fits a ball with this diameter.

If this does not hold the algorithm can return a wrong result.

• domain: A tuple in the form (xmin, xmax, ymin, ymax) which defines a section Ω

for which the amoeba $\mathcal{A}(f)$ is computed. This domain has to be such that the intersection Ω ∩ $\mathcal{A}(f)$ still captures the correct topology of $\mathcal{A}(f)$.

• grid: Based on minimal_component_size and domain a grid can be computed automatically.

This can also be overwritten with this option.

• membership_options=[MembershipTestOptions](@ref)(): Options for the membership test.
• nsamples=1024: For the computation we numerically evaluate intergrals. This is the number

of sample points used in the computation.

spine(f::MP.AbstractPolynomial, A::Bitmap2D; nsamples=1024)

Compute the spine based on the given amoeba approximation A.

spine(A::PolygonalAmoeba)

The spine which was used to compute A.

A Spine2D represents the spine of a two-dimensional amoeba. This also contains informations about the components of the complement and the approximation used to obtain this spine.

Represents a component of the complement of the amoeba. It contains information about the order of the component, whether it is bounded and one point out of the component.

components_complement(S::Spine2D)

Get the ComponentComplement of the spine S.

ronkin_polynomial(S::Spine2D)

Get the tropical polynomial which defines this hypersurface.

hypersurface(S::Spine2D)

Get the tropical curve which describes the spine.

amoeba_approximation(S::Spine2D)

Get the approximated amoeba used to compute the spine.