documentation:release:4.11:tropical:hypersurface

Available versions of this document: latest release, release 4.12, release 4.11, release 4.10, release 4.9, release 4.8, release 4.7, release 4.6, release 4.5, release 4.4, release 4.3, release 4.2, release 4.1, release 4.0, release 3.6, release 3.5, nightly master

Reference documentation for older polymake versions: release 3.4, release 3.3, release 3.2

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.

• documentation/release/4.11/tropical/hypersurface.txt