Reference

Reference

Grid

PolynomialAmoebas.Grid2D — Type.

A structure representing a 2D Grid.

PolynomialAmoebas.Grid3D — Type.

A structure representing a 3D Grid.

Membership test

PolynomialAmoebas.membershiptest — Function.
membershiptest(F::AbstractFiber{T}, w:, options=MembershipTestOptions()) where T

Check wether the point w is contained in the fiber F. The tests uses Newton's method with configuration provided by options. Returns a MembershipTestResult.

PolynomialAmoebas.MembershipTestOptions — Type.
MembershipTestOptions(iterations=50, ntries=30, tol=1e-7)

Details

  • iterations The maximal number of iterations in one try.
  • ntries The number of times Newton's method will be tried
  • tol The desired accuracy when Newton's method converges.
PolynomialAmoebas.MembershipTestResult — Type.
MembershipTestResult{T, N}

Fields

  • successfull::Bool
  • tries::Int The number of times Newton's method was started
  • converged_iterations::Int Number iterations needed in a successfull iteration. If the test was not successfull imax is the maximal number of iterations.
  • startvalue::SVector{N, T} The start value of the successfull iteration. Otherwise a random value.
  • solution::SVector{N, T} The solution value of the successfull iteration. Otherwise a random value.

Newton polygon

PolynomialAmoebas.NewtonPolygon — Type.
NewtonPolygon

Representation of a Newton polygon of a polynomial.

Fields

  • polynomial::P: the polynomial from which the NewtonPolygon was constructed.
  • lattices::Vector{SVector{2,Int}}: the support of the polynomial p.
  • vertices::Vector{Int}: the vertex indicdes of the Newton polygon.
  • facets::Vector{SVector{2,Int}}: the facets of the Newton polygon, i.e. it's convex hull.
  • subdivision::Vector{SVector{3,Int}}: the simplices of the regular subdivision of the

Newton polygon

PolynomialAmoebas.newtonpolygon — Function.
newtonpolygon(p::AbstractPolynomial; lowerhull=false)

Construct a NewtonPolygon from the support of the polynomial p. lowerhull indicates whether the lower or upper convex hull should be used to construct the regular subdivision of the Newton polygon.

Fibers

PolynomialAmoebas.AmoebaFiber2D — Type.

An AmoebaFiber2D is a representation of the fiber of the amoeba $\mathcal{A}_f$ at $(w_1,w_2)$ where f is a bivariate polynomial.

PolynomialAmoebas.AmoebaFiber3D — Type.

An AmoebaFiber3D is a representation of the fiber of the amoeba $\mathcal{A}_f$ at $(w_1, w_2, w_3)$ where f is a trivariate polynomial.

PolynomialAmoebas.ContourFiber2D — Type.

An ContourFiber2D is a representation of the fiber of the contour of the amoeba $\mathcal{A}^{\prime}_f$ at $(w_1, w_2)$ along a onedimensional affine subspace where f is a bivariate polynomial.

PolynomialAmoebas.CoamoebaFiber2D — Type.

An CoamoebaFiber2D is a representation of the fiber of the coamoeba $\mathcal{A}^{\prime}_f$ at $(θ_1, θ_2)$ where f is a bivariate polynomial.

PolynomialAmoebas.CoamoebaFiber3D — Type.
CoamoebaFiber3D(f::AbstractPolynomial, θ=(0., 0., 0.))

Construct the fiber CoamoebaFiber3D of the trivariate polynomial f at θ.

PolynomialAmoebas.ImaginaryFiber2D — Type.
ImaginaryFiber2D(f, y=(0., 0.))

Construct the fiber ImaginaryFiber2D of the bivariate polynomial f at y.

PolynomialAmoebas.ImaginaryFiber3D — Type.
ImaginaryFiber2D(f, y=(0., 0., 0.))

Construct the fiber ImaginaryFiber3D of the trivariate polynomial f at y.

Tropical geometry

PolynomialAmoebas.amoeba_carcase_domain_heuristic — Function.
amoeba_carcase_domain_heuristic(f; factor=1.5, aspect_ratio=:default)

Compute the boundary (xmin, xmax, ymin, ymax) of domain Ω such that Ω ∩ Amoeba(f) is probably the carcase of Amoeba(f).

PolynomialAmoebas.archimedean_tropical_curve — Function.
archimedean_tropical_curve(f)

Compute the curve associated to the Archimedean tropical polynomial of f.

PolynomialAmoebas.TropicalCurve — Type.

A TropicalCurve is a tropical hypersurface of a bivariate polynomial.

PolynomialAmoebas.vertices — Method.
vertices(tropicalcurve)

The vertices of the tropical curve.

PolynomialAmoebas.segments — Method.
segments(tropicalcurve)

The segments of the tropical curve. Each segment is a SVector{2} with start and end coordinates.

PolynomialAmoebas.halfrays — Method.
halfrays(tropicalcurve)

The halfrays of the tropical curve. Each halfray is a tuple (coordinate, direction).