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- ItemSupertropical linear algebra(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2010) Izhakian, Zur; Knebusch, Manfred; Rowen, LouisThe 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.
- 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.
- ItemSupertropical matrix algebra III: Powers of matrices and generalized eigenspaces(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2010) Izhakian, Zur; Rowen, LouisWe investigate powers of supertropical matrices, with special attention to the role of the co- e±cients of the supertropical characteristic polynomial (especially the supertropical trace) in controlling the rank of a power of a matrix. This leads to a Jordan-type decomposition of supertropical matrices, together with a generalized eigenspace decomposition of a power of an arbitrary supertropical matrix.