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- ItemSupertropical quadratic forms I(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2013) Izhakian, Zur; Knebusch, Manfred; Rowen, LouisWe 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.
- ItemMonoid valuations and value ordered supervaluations(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2011) Izhakian, Zur; Knebusch, Manfred; Rowen, LouisWe 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.
- ItemDominance and transmissions in supertropical valuation theory(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2011) Izhakian, Zur; Knebusch, Manfred; Rowen, LouisThis 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.
- ItemNew representations of matroids and generalizations(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2011) Izhakian, Zur; Rhodes, JohnWe extend the notion of matroid representations by matrices over fields by considering new representations of matroids by matrices over finite semirings, more precisely over the boolean and the superboolean semirings. This idea of representations is naturally generalized to include hereditary collections (also known as abstract simplicial complexes). We show that a matroid that can be directly decomposed as matroids, each of which is representable over a field, has a boolean representation, and more generally that any arbitrary hereditary collection is superboolean-representable.