
D.13.3 realizationMatroids_lib
 Library:
 realizationMatroids.lib
 Purpose:
 Deciding Relative Realizability for Tropical Fan Curves in 2Dimensional Matroidal Fans
 Authors:
 Anna Lena Winstel, winstel@mathematik.unikl.de
 Overview:
 In tropical geometry, one question to ask is the following: given a onedimensional balanced polyhedral fan C which is set theoretically contained in the tropicalization trop(Y) of an algebraic variety Y, does there exist a curve X in Y such that trop(X) = C? This equality of C and trop(X) denotes an equality of both, the fans trop(X) and C and their weights on the maximal cones. The relative realization space of C with respect to Y is the space of all algebraic curves in Y which tropicalize to C.
This library provides procedures deciding relative realizability for tropical fan curves, i.e. onedimensional weighted balanced polyhedral fans, contained in twodimensional matroidal fans trop(Y) where Y is a projective plane.
 Notation:
 If Y is a projective plane in (n1)dimensional projective space, we consider trop(Y) in R^n/<1>. Moreover, for the relative realization space of C with respect to Y we only consider algebraic curves of degree deg(C) in Y which tropicalize to C.
Procedures:
D.13.3.1 realizationDim   For a given tropical fan curve C in trop(Y), where Y = V(I) is a projective plane, this routine returns the dimension of the relative realization space of C with respect to Y, that is the space of all algebraic curves of degree deg(C) in Y which tropicalize to C. If the realization space is empty, the output is set to 1. 
D.13.3.2 irrRealizationDim   This routine returns the dimension of the irreducible relative realization space of the tropical fan curve C with respect to Y = V(I), that is the space of all irreducible algebraic curves of degree deg(C) in Y which tropicalize to C. If the irreducible relative realization space is empty, the output is set to 1. 
D.13.3.3 realizationDimPoly   If C is a tropical fan curve contained in the tropicalization trop(Y) of the projective plane Y = V(I) such that the relative realization space M of C is nonempty, this routine returns the tuple (dim(M),f) where f is an example of a homogeneous polynomial of degree deg(C) cutting out a curve X in Y which tropicalizes to C. If M is empty, the output is set to 1. 
