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BigObject Hypersurface<Addition>
from application 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 MONOMIALS
is never checked.
 Type Parameters:
 derived from:
 Example:
The following yields a tropical plane conic.
> $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]);
 Permutations:
 TermPerm:
permuting
MONOMIALS
andCOEFFICIENTS
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.

DUAL_SUBDIVISION
Subdivision of the Newton polytope dual to the tropical hypersurface. The vertices of this PolyhedralComplex are the nonredundant
MONOMIALS
. Type:

PATCHWORK
This encodes a patchworking structure on a tropical hypersurface. Its lone input property is
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:
 Example:
> $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]);
no category

COEFFICIENTS
Coefficients of the homogeneous tropical polynomial
POLYNOMIAL
. Each entry corresponds to one of the terms inPOLYNOMIAL
. The order is compatible with the order ofMONOMIALS
. Type:
Vector<TropicalNumber<Addition,Rational>>

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, 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]]>''

MONOMIALS
Exponent vectors of the homogeneous tropical polynomial
POLYNOMIAL
. Each row corresponds to one of the monomials inPOLYNOMIAL
, each column to a variable. Type:

POLYNOMIAL
Homogeneous tropical polynomial defining the hypersurface. Note: The homogeneity of the polynomial is never checked.
 Type:
Polynomial<TropicalNumber<Addition,Rational>,Int>

REDUNDANT_MONOMIALS
 Type:

REGIONS
Connected components of the complement. Rows correspond to facets of the
DOME
, i.e. nonredundantMONOMIALS
, columns correspond toVERTICES
. Type:
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.

dual_subdivision()
Returns
DUAL_SUBDIVISION
; backward compatibility. Returns: