### Abstract

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 |
---|---|

Pages (from-to) | 2222-2286 |

Number of pages | 65 |

Journal | Advances in Mathematics |

Volume | 225 |

Issue number | 4 |

Early online date | 1 Jun 2010 |

DOIs | |

Publication status | Published - 10 Nov 2010 |

### Fingerprint

### Keywords

- polynomials
- tropical geometry
- max-plus algebra
- valuations
- valued monoids
- semirings
- supertropical algebra
- supertropical semirings
- ideals
- ghost ideals
- prime ideals
- Nullstellesatz
- polynomial factorization

### Cite this

*Advances in Mathematics*,

*225*(4), 2222-2286. https://doi.org/10.1016/j.aim.2010.04.007

**Supertropical algebra.** / Izhakian, Zur; Rowen, Louis .

Research output: Contribution to journal › Article

*Advances in Mathematics*, vol. 225, no. 4, pp. 2222-2286. https://doi.org/10.1016/j.aim.2010.04.007

}

TY - JOUR

T1 - Supertropical algebra

AU - Izhakian, Zur

AU - Rowen, Louis

PY - 2010/11/10

Y1 - 2010/11/10

N2 - 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.

AB - 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.

KW - polynomials

KW - tropical geometry

KW - max-plus algebra

KW - valuations

KW - valued monoids

KW - semirings

KW - supertropical algebra

KW - supertropical semirings

KW - ideals

KW - ghost ideals

KW - prime ideals

KW - Nullstellesatz

KW - polynomial factorization

U2 - 10.1016/j.aim.2010.04.007

DO - 10.1016/j.aim.2010.04.007

M3 - Article

VL - 225

SP - 2222

EP - 2286

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

IS - 4

ER -