## Abstract

We develop the algebraic polynomial theory for "supertropical algebra," as initiated earlier over the real numbers by the first author. The main innovation there was the introduction of "ghost elements," which also play the key role in our structure theory. Here, we work somewhat more generally over an ordered monoid, and develop a theory which contains the analogs of several basic theorems of classical commutative algebra. This structure enables one to develop a Zariski-type algebraic geometric approach to tropical geometry, viewing tropical varieties as sets of roots of (supertropical) polynomials, leading to an analog of the Hilbert Nullstellensatz.Particular attention is paid to factorization of polynomials. In one indeterminate, any polynomial can be factored into linear and quadratic factors, and although unique factorization may fail, there is a "preferred" factorization that is explained both geometrically and algebraically. The failure of unique factorization in several indeterminates is explained by geometric phenomena described in the paper.

Original language | English |
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Pages (from-to) | 2222-2286 |

Number of pages | 65 |

Journal | Advances in Mathematics |

Volume | 225 |

Issue number | 4 |

DOIs | |

State | Published - Nov 2010 |

### Bibliographical note

Funding Information:* Corresponding author at: Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel. E-mail addresses: [email protected], [email protected] (Z. Izhakian), [email protected] (L. Rowen). 1 The first author was supported by the Chateaubriand scientific post-doctorate fellowships, Ministry of Science, French Government, 2007–2008.

## Keywords

- Ghost ideals
- Ideals
- Max-plus algebra
- Nullstellensatz
- Polynomial factorization
- Polynomials
- Primary
- Prime ideals
- Secondary
- Semirings
- Supertropical algebra
- Supertropical semirings
- Tropical geometry
- Valuations
- Valued monoids