Supertropical algebra

By Zur Izhakian, Louis H. Rowen. 2008

127 citations recorded by the source corpus.

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 unique factorization holds in a certain sense. On the other hand, the failure of unique factorization in several indeterminates is explained by geometric phenomena described in the paper.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/25f15556e65d973e474a339fc99e4730de9206f0

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Citations in the corpus

Supertropical matrix algebra (2008, background): “a+b is a ghost element, although the converse might fail. For tropical applications, we focus on the tangible elements, which in this paper are deﬁned as T = R \G 0; they are deﬁned more generally in [11]. One diﬀerence with [11] is that here we do not require semirings to be commutative, since we d”
The Tropical Rank of a Tropical Matrix (2006, methodology): “The main goal of this paper is a further development of the basics of tropical matrix algebra over the extended tropical semiring, (T,⊕,⊙), as has been presented in [5]; we also use some of the terminology used in [6].”
Tropical Cramer Determinants Revisited (2013, methodology): “We obtain existence or uniqueness results, which extend or refine earlier results of Gondran and Minoux (1978), Plus (1990), Gaubert (1992), Richter-Gebert, Sturmfels and Theobald (2005) and Izhakian and Rowen (2009).”
Supertropical matrix algebra II: Solving tropical equations (2009, background): “We recall that this work is in the environment of a semiring with ghosts [5], which is a triple (R,G , ν), where R is a semiring with zero element, R (often identified in the examples with −∞, as indicated below), and G = G ∪ { R} is a semiring ideal, called the ghost ideal, together with an idempot”
Algebraic Geometry Over Hyperrings (2015, background): “Note that supertropical algebras by Z.Izhakian and L.Rowen (cf. [21], [22]) and blueprints by O.Lorscheid (cf. [28], [29]) provide natural algebraic framework for tropical geometry and have certain connections to hyperrings.”