## Abstract

A tropical curve Γ is a metric graph with possibly unbounded edges, and tropical rational functions are continuous piecewise linear functions with integer slopes. We define the complete linear system |D| of a divisor D on a tropical curve Γ analogously to the classical counterpart. We investigate the structure of |D| as a cell complex and show that linear systems are quotients of tropical modules, finitely generated by vertices of the cell complex. Using a finite set of generators, |D| defines a map from Γ to a tropical projective space, and the image can be modified to a tropical curve of degree equal to deg(D). The tropical convex hull of the image realizes the linear system |D| as a polyhedral complex.

Original language | English (US) |
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Title of host publication | FPSAC'10 - 22nd International Conference on Formal Power Series and Algebraic Combinatorics |

Pages | 295-306 |

Number of pages | 12 |

State | Published - Dec 1 2010 |

Event | 22nd International Conference on Formal Power Series and Algebraic Combinatorics, FPSAC'10 - San Francisco, CA, United States Duration: Aug 2 2010 → Aug 6 2010 |

### Other

Other | 22nd International Conference on Formal Power Series and Algebraic Combinatorics, FPSAC'10 |
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Country | United States |

City | San Francisco, CA |

Period | 8/2/10 → 8/6/10 |

## Keywords

- Canonical embedding
- Chip-firing games
- Divisors
- Linear systems
- Tropical convexity
- Tropical curves