Supertropical linear algebra

TL;DR

This paper develops the algebraic framework of supertropical linear algebra, introducing concepts like bases, dual spaces, and bilinear forms, and establishing foundational theorems within this novel mathematical setting.

Contribution

It introduces the supertropical vector space theory, compares different bases, and extends classical linear algebra results to the supertropical context.

Findings

Existence and uniqueness of s-base up to permutation and scalar multiplication.

Development of supertropical bilinear forms and Gram matrix.

Connection of supertropical dependence with the supertropical Artin theorem.

Abstract

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 identified 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.