Supertropical algebra

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.

Completions, Reversals, and Duality for Tropical Varieties

“…Specifically, if f i are the monomials of f , then f is a factor of the product i =j (f i + f j ), as was seen in [6,Theorem 12.4]. On the other hand, Theorem 0.1 has a positive geometric interpretation -Every tropical variety W can be "completed" to a variety P(W ) comprised of various k-dimensional planes, which in turn can be decomposed into a union of varieties that can be interpreted via (1).…”

“…Factorization in polynomials over Ì max is notoriously difficult, cf. [6,7]. One reason is that different polynomials in Ì[λ 1 , .…”

Supertropical matrix algebra

“…So we also need to pinpoint some of those properties that are preserved in such matrix semirings. Our underlying structure is a semiring with ghosts, which we recall from [13] is a triplet (R, G ¼ , ν), where R is a semiring with a unit element ½ R and with zero element ¼ R (satisfying ¼ R r = r ¼ R = ¼ R for every r ∈ R, and often identified in the examples with −∞, as indicated below), G ¼ = G ∪ {¼ R } is a semiring ideal called the ghost ideal, and ν : R → G ¼ , called the ghost map, is an idempotent semiring homomorphism (i.e., which preserves multiplication as well as addition).…”

“…In [13], the abstract foundations of supertropical algebra were set forth, including the concept of a supertropical domain and supertropical semifield. The motivation was to overcome the difficulties inherent in studying polynomials over the max-plus algebra, by providing an algebraic structure that encompasses the max-plus algebra, thereby permitting a thorough study of polynomials and their roots and a direct algebraic-geometric development of tropical geometry.…”

Questions inspired by Mikael Passare’s mathematics

“…In his answer of 2010-03-29, Mikael directed me to the preprint by Izhakian and Rowen [13], published as [14]. (Perhaps the paper by Izhakian [12] is easier to start with.)…”