# Supertropical linear algebra

@article{Izhakian2010SupertropicalLA, title={Supertropical linear algebra}, author={Zur Izhakian and Manfred Knebusch and Louis H. Rowen}, journal={arXiv: Commutative Algebra}, year={2010} }

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…

## 34 Citations

Supertropical Quadratic Forms II

- Mathematics
- 2015

This article is a sequel of [4], where we introduced quadratic forms on a module V over a supertropical semiring R and analysed the set of bilinear companions of a quadratic form q : V → R in case…

Layered Tropical Mathematics

- Mathematics
- 2009

Generalizing supertropical algebras, we present a "layered" structure, "sorted" by a semiring which permits varying ghost layers, and indicate how it is more amenable than the "standard"…

ELT Linear Algebra

- Mathematics
- 2016

Exploded layered tropical (ELT) algebra is an extension of tropical algebra with a structure of layers. These layers allow us to use classical algebraic results in order to easily prove analogous…

Dual spaces and bilinear forms in supertropical linear algebra

- Mathematics
- 2012

Continuing 4, this article investigates finer points of supertropical vector spaces, including dual bases and bilinear forms, with supertropical versions of standard classical results such as the…

On pseudo-inverses of matrices and their characteristic polynomials in supertropical algebra☆

- Mathematics
- 2013

Abstract The only invertible matrices in tropical algebra are diagonal matrices, permutation matrices and their products. However, the pseudo-inverse A ∇ , defined as 1 det ( A ) adj ( A ) , with…

Supertropical matrix algebra III: Powers of matrices and their supertropical eigenvalues☆

- Mathematics
- 2011

Abstract We investigate powers of supertropical matrices, with special attention to the role of the coefficients of the supertropical characteristic polynomial (especially the supertropical trace) in…

Supertropical monoids: Basics and canonical factorization

- Mathematics
- 2013

Abstract A supertropical monoid is a monoid U together with a projection onto a totally ordered submonoid e U (where e ∈ U is idempotent). Supertropical monoids are slightly more general than the…

Determining Tropical Hypersurfaces

- Mathematics
- 2015

We consider the question of when points in tropical affine space uniquely determine a tropical hypersurface. We introduce a notion of multiplicity of points so that this question may be meaningful…

Supertropical Matrix Algebra III : Powers of Matrices and Generalized Eigenspaces

- Mathematics
- 2010

We investigate powers of supertropical matrices, with special attention to the role of the coefficients of the supertropical characteristic polynomial (especially the supertropical trace) in…

Commutative ν-Algebra and Supertropical Algebraic Geometry

- Mathematics
- 2019

This paper lays out a foundation for a theory of supertropical algebraic geometry, relying on commutative $\nu$-algebra. To this end, the paper introduces $\mathfrak{q}$-congruences, carried over…

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Supertropical matrix algebra III: Powers of matrices and their supertropical eigenvalues☆

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Abstract We investigate powers of supertropical matrices, with special attention to the role of the coefficients of the supertropical characteristic polynomial (especially the supertropical trace) in…

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Supertropical Matrix Algebra III : Powers of Matrices and Generalized Eigenspaces

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We investigate powers of supertropical matrices, with special attention to the role of the coefficients of the supertropical characteristic polynomial (especially the supertropical trace) in…

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