Supertropical algebra

Zur Izhakian, Louis Rowen

Research output: Contribution to journalArticle

58 Citations (Scopus)

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 languageEnglish
Pages (from-to)2222-2286
Number of pages65
JournalAdvances in Mathematics
Volume225
Issue number4
Early online date1 Jun 2010
DOIs
Publication statusPublished - 10 Nov 2010

Keywords

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

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