====== BigObject Hypersurface ====== //from application [[..:tropical|tropical]]//\\ \\ This is a special instance of a Cycle: It is the tropical locus of a homogeneous polynomial over the tropical numbers. Note: Homogeneity of the polynomial is never checked. ? Type Parameters: :: ''Addition'': Mode of tropical addition, must be ''[[..:common#Min |Min]]'' or ''[[..:common#Max |Max]]''. There is on purpose no default value for it. ? derived from: : ''[[..:tropical:Cycle |Cycle]]'' ? Permutations: : ? TermPerm: :: permuting ''[[..:tropical:Hypersurface#MONOMIALS |MONOMIALS]]'' and ''[[..:tropical:Hypersurface#COEFFICIENTS |COEFFICIENTS]]'' ===== Properties ===== ==== Combinatorics ==== These properties capture combinatorial information of the object. Combinatorial properties only depend on combinatorial data of the object like, e.g., the face lattice. ---- {{anchor:dual_subdivision:}} ? **''DUAL_SUBDIVISION''** :: Subdivision of the Newton polytope dual to the tropical hypersurface. The vertices of this PolyhedralComplex are the non-redundant ''[[..:tropical:Hypersurface#MONOMIALS |MONOMIALS]]''. ? Type: :''[[..:fan:SubdivisionOfPoints |SubdivisionOfPoints]]<[[..:common#Rational |Rational]]>'' ---- {{anchor:patchwork:}} ? **''PATCHWORK''** :: This encodes a patchworking structure on the hypersurface. Its lone input property is ''[[..:tropical:Patchwork#SIGNS |SIGNS]]'' ? Type: :''[[..:tropical:Patchwork |Patchwork]]'' ---- ==== no category ==== {{anchor:coefficients:}} ? **''COEFFICIENTS''** :: Coefficients of the homogeneous tropical polynomial ''[[..:tropical:Hypersurface#POLYNOMIAL |POLYNOMIAL]]''. Each entry corresponds to one of the terms in ''[[..:tropical:Hypersurface#POLYNOMIAL |POLYNOMIAL]]''. The order is compatible with the order of ''[[..:tropical:Hypersurface#MONOMIALS |MONOMIALS]]''. ? Type: :''[[..:common#Vector |Vector]]<[[..:common#TropicalNumber |TropicalNumber]]>%%'' ---- {{anchor:dome:}} ? **''DOME''** :: The __dome__ of a (inhomogeneous) tropical polynomial \(F:\mathbb R^d\to\mathbb R\) (and the corresponding affine tropical hypersurface) is the set \[D(F)=\left\{(p,s)\in\mathbb R^{d+1}\mid p\in\mathbb R^d, s\in\mathbb R, s \oplus F(p) = s\right\}\]. It is an unbounded convex polyhedron, c.f. > Michael Joswig, [[http://page.math.tu-berlin.de/~joswig/etc/|Essentials of Tropical Combinatorics]], Chapter 1. .. For a projective tropical hypersurface, the __dome__ is the intersection of the dome for the affine case with the hyperplane at height 1. Note: To account for negative exponents, the dome may have been repositioned by multiplying the original polynomial with a suitable monomial. ? Type: :''[[..:polytope:Polytope |Polytope]]<[[..:common#Rational |Rational]]>'' ---- {{anchor:monomials:}} ? **''MONOMIALS''** :: Exponent vectors of the homogeneous tropical polynomial ''[[..:tropical:Hypersurface#POLYNOMIAL |POLYNOMIAL]]''. Each row corresponds to one of the monomials in ''[[..:tropical:Hypersurface#POLYNOMIAL |POLYNOMIAL]]'', each column to a variable. ? Type: :''[[..:common#Matrix |Matrix]]<[[..:common#Int |Int]],[[..:common#NonSymmetric |NonSymmetric]]>'' ---- {{anchor:polynomial:}} ? **''POLYNOMIAL''** :: Homogeneous tropical polynomial defining the hypersurface. Note: Homogeneity of the polynomial is never checked. ? Type: :''[[..:common#Polynomial |Polynomial]]<[[..:common#TropicalNumber |TropicalNumber]],[[..:common#Int |Int]]>'' ---- {{anchor:redundant_monomials:}} ? **''REDUNDANT_MONOMIALS''** :: Indices of ''[[..:tropical:Hypersurface#MONOMIALS |MONOMIALS]]'' which do not define facets of the ''[[..:tropical:Hypersurface#DOME |DOME]]''. ? Type: :''[[..:common#Set |Set]]<[[..:common#Int |Int]]>'' ---- {{anchor:regions:}} ? **''REGIONS''** :: Connected components of the complement. Rows correspond to facets of the ''[[..:tropical:Hypersurface#DOME |DOME]]'', i.e. non-redundant ''[[..:tropical:Hypersurface#MONOMIALS |MONOMIALS]]'' , columns correspond to ''[[..:fan:PolyhedralComplex#VERTICES |VERTICES]]''. ? Type: :''[[..:common#IncidenceMatrix |IncidenceMatrix]]<[[..:common#NonSymmetric |NonSymmetric]]>'' ---- ===== Methods ===== ==== Combinatorics ==== These methods capture combinatorial information of the object. Combinatorial properties only depend on combinatorial data of the object like, e.g., the face lattice. ---- {{anchor:dual_subdivision:}} ? **''dual_subdivision()''** :: Returns ''[[..:tropical:Hypersurface#DUAL_SUBDIVISION |DUAL_SUBDIVISION]]''; backward compatibility. ? Returns: :''[[..:fan:SubdivisionOfPoints |SubdivisionOfPoints]]'' ---- ==== Topology ==== The following methods compute topological invariants. ---- {{anchor:genus:}} ? **''GENUS''** :: The topological genus of a onedimensional hypersurface, i.e. the number of interior lattice points that occur in the dual subdivision. ----