Quadratic and Symmetric Bilinear Forms on Modules with Unique Base Over a Semiring

TL;DR

This paper investigates quadratic forms on free modules with unique bases in tropical algebra, proving an analog of Witt's Cancellation Theorem and analyzing tensor products of indecomposable modules.

Contribution

It introduces a Witt's Cancellation Theorem analog for modules over semirings and characterizes indecomposability of tensor products of modules.

Findings

Proves an analog of Witt's Cancellation Theorem for tropical modules.

Shows tensor product of indecomposable modules is mostly indecomposable.

Identifies a specific case where tensor products decompose into two components.

Abstract

We study quadratic forms on free modules with unique base, the situation that arises in tropical algebra, and prove the analog of Witt's Cancellation Theorem. Also, the tensor product of an indecomposable bilinear module $(U, \gamma)$ with an indecomposable quadratic module $(V,q) $ is indecomposable, with the exception of one case, where two indecomposable components arise.