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:

Addition: The tropical addition. Warning: There is NO default for this, you have to choose either Max or Min.

derived from:

Cycle

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 and COEFFICIENTS

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 non-redundant MONOMIALS.

Type:

SubdivisionOfPoints<Rational>

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:

Patchwork

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]);

COEFFICIENTS

Coefficients of the homogeneous tropical polynomial POLYNOMIAL. Each entry corresponds to one of the terms in POLYNOMIAL. The order is compatible with the order of MONOMIALS.

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 in POLYNOMIAL, each column to a variable.

Type:

Matrix<Int,NonSymmetric>

POLYNOMIAL

Homogeneous tropical polynomial defining the hypersurface. Note: The homogeneity of the polynomial is never checked.

Type:

Polynomial<TropicalNumber<Addition,Rational>,Int>

REDUNDANT_MONOMIALS

Indices of MONOMIALS which do not define facets of the DOME.

Type:

Set<Int>

REGIONS

Connected components of the complement. Rows correspond to facets of the DOME, i.e. non-redundant MONOMIALS , columns correspond to VERTICES.

Type:

IncidenceMatrix<NonSymmetric>

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:

SubdivisionOfPoints

The following methods compute topological invariants.

GENUS

The topological genus of a onedimensional hypersurface, i.e. the number of interior lattice points that occur in the dual subdivision.