2011, Izhakian, Zur, Knebusch, Manfred, Rowen, Louis

We complement two papers on supertropical valuation theory ([IKR1], [IKR2]) by providing natural examples of m-valuations (= monoid valuations), after that of supervaluations and transmissions between them. The supervaluations discussed have values in totally ordered supertropical semirings, and the transmissions discussed respect the orderings. Basics of a theory of such semirings and transmissions are developed as far as needed.

2011, Izhakian, Zur, Knebusch, Manfred, Rowen, Louis

This paper is a sequel of [IKR1], where we defined supervaluations on a commutative ring R and studied a dominance relation Φ>=v between supervaluations φ and υ on R, aiming at an enrichment of the algebraic tool box for use in tropical geometry. A supervaluation φ:R→U is a multiplicative map from R to a supertropical semiring U, cf. [IR1], [IR2], [IKR1], with further properties, which mean that φ is a sort of refinement, or covering, of an m-valuation (= monoid valuation) υ:R→M. In the most important case, that R is a ring, m-valuations constitute a mild generalization of valuations in the sense of Bourbaki [B], while φ>=υ means that υ:R→V is a sort of coarsening of the supervaluation φ. If φ(R) generates the semiring U, then φ>=υ if there exists a "transmission" α:U→V with φ=α∘φ. Transmissions are multiplicative maps with further properties, cf. [IKR1, §55]. Every semiring homomorphism α:U→V is a transmission, but there are others which lack additivity, and this causes a major difficulty. In the main body of the paper we study surjective transmissions via equivalence relations on supertropical semirings, often much more complicated than congruences by ideals in usual commutative algebra.

2013, Izhakian, Zur, Knebusch, Manfred, Rowen, Louis

We initiate the theory of a quadratic form q over a semiring R. As customary, one can write q(x+y)=q(x)+q(y)+b(x,y), where b is a companion bilinear form. But in contrast to the ring-theoretic case, the companion bilinear form need not be uniquely defined. Nevertheless, q can always be written as a sum of quadratic forms q=κ+ρ, where κ is quasilinear in the sense that κ(x+y)=κ(x)+κ(y), and ρ is rigid in the sense that it has a unique companion. In case that R is a supersemifield (cf. Definition 4.1 below) and q is defined on a free R-module, we obtain an explicit classification of these decompositions q=κ+ρ and of all companions b of q. As an application to tropical geometry, given a quadratic form q:V→R on a free module V over a commutative ring R and a supervaluation ρ: R→U with values in a supertropical semiring [5], we define - after choosing a base L=(vi|i∈I) of V- a quadratic form qφ:U(I)→U on the free module U(I) over the semiring U. The analysis of quadratic forms over a supertropical semiring enables one to measure the “position” of q with respect to L via φ.

2010, Izhakian, Zur, Knebusch, Manfred, Rowen, Louis

The objective of this paper is to lay out the algebraic theory of supertropical vector spaces and linear algebra, utilizing the key antisymmetric relation of \ghost surpasses." Special attention is paid to the various notions of \base," which include d-base and s-base, and these are compared to other treatments in the tropical theory. Whereas the number of elements in a d-base may vary according to the d-base, it is shown that when an s-base exists, it is unique up to permutation and multiplication by scalars, and can be identi¯ed with a set of \critical" elements. Linear functionals and the dual space are also studied, leading to supertropical bilinear forms and a supertropical version of the Gram matrix, including its connection to linear dependence, as well as a supertropical version of a theorem of Artin.

2010, Izhakian, Zur, Knebusch, Manfred, Rowen, Louis

We interpret a valuation v on a ring R as a map v : R ! M into a so called bipotent semiring M (the usual max-plus setting), and then de¯ne a supervaluation ' as a suitable map into a supertropical semiring U with ghost ideal M (cf. [IR1], [IR2]) covering v via the ghost map U ! M. The set Cov(v) of all supervaluations covering v has a natural ordering which makes it a complete lattice. In the case that R is a field, hence for v a Krull valuation, we give a complete explicit description of Cov(v). The theory of supertropical semirings and supervaluations aims for an algebra fitting the needs of tropical geometry better than the usual max-plus setting. We illustrate this by giving a supertropical version of Kapranov's lemma.