TY - JOUR

T1 - Supertropical algebra

AU - Izhakian, Zur

AU - Rowen, Louis

N1 - 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.

PY - 2010/11

Y1 - 2010/11

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 - Ghost ideals

KW - Ideals

KW - Max-plus algebra

KW - Nullstellensatz

KW - Polynomial factorization

KW - Polynomials

KW - Primary

KW - Prime ideals

KW - Secondary

KW - Semirings

KW - Supertropical algebra

KW - Supertropical semirings

KW - Tropical geometry

KW - Valuations

KW - Valued monoids

UR - http://www.scopus.com/inward/record.url?scp=77956177039&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2010.04.007

DO - 10.1016/j.aim.2010.04.007

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AN - SCOPUS:77956177039

SN - 0001-8708

VL - 225

SP - 2222

EP - 2286

JO - Advances in Mathematics

JF - Advances in Mathematics

IS - 4

ER -