====== BigObject Hypersurface<Addition> ======
//from application [[..:tropical|tropical]]//\\
\\
 Tropical hypersurface in the tropical projective torus R^d/R1. 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 ''[[..:tropical:Hypersurface#MONOMIALS |MONOMIALS]]'' is never checked.

  ? Type Parameters:
  :: ''Addition'': The tropical addition. Warning: There is NO default for this, you have to choose either ''[[..:common#Max |Max]]'' or ''[[..:common#Min |Min]]''.
  ? derived from:
  : ''[[..:tropical:Cycle |Cycle]]''
  ? Example:
  :: The following yields a tropical plane conic.
  :: <code perl> > $C=new Hypersurface<Min>(MONOMIALS=>[ [2,0,0],[1,1,0],[0,2,0],[1,0,1],[0,1,1],[0,0,2] ], COEFFICIENTS=>[6,5,6,5,5,7]);
</code>
  ? 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.
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{{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 a tropical hypersurface. Its lone input property is ''[[..:tropical:Patchwork#SIGNS |SIGNS]]'' - a sign distribution on the vertices of the induced regular subdivision of the corresponding Newton polytope. As it is a multiple subobject, you can create multiple patchworking structures (for different sign distributions) on the same tropical hypersurface object.
    ? Type:
    :''[[..:tropical:Patchwork |Patchwork]]''
    ? Example:
    :: <code perl> > $h = new tropical::Hypersurface<Max>(POLYNOMIAL=>toTropicalPolynomial("max(a,b,c)"));
 > $p1 = $h->PATCHWORK(SIGNS=>[0,1,0]);
 > $p2 = $h->PATCHWORK(SIGNS=>[1,1,1]);
</code>


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==== 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]]<Addition,[[..:common#Rational |Rational]]%%>>%%''


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{{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]]>''


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{{anchor:polynomial:}}
  ?  **''POLYNOMIAL''**
  :: Homogeneous tropical polynomial defining the hypersurface. Note: The homogeneity of the polynomial is never checked.
    ? Type:
    :''[[..:common#Polynomial |Polynomial]]<[[..:common#TropicalNumber |TropicalNumber]]<Addition,[[..:common#Rational |Rational]]>,[[..: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]]>''


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===== 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]]''


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==== Topology ====
 The following methods compute topological invariants.
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{{anchor:genus:}}
  ?  **''GENUS''**
  :: The topological genus of a onedimensional hypersurface, i.e. the number of interior lattice points that occur in the dual subdivision.


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