Subordinate quadratic forms and isometric maps over semirings

TL;DR

This paper advances the theory of quadratic forms over semirings, focusing on isometric maps that enable the transfer and preservation of quadratic form properties in tropical and supertropical algebra contexts.

Contribution

It introduces the concept of subordinate quadratic forms and explores isometric maps that facilitate lifting and pushing down quadratic forms across modules over semirings.

Findings

Isometric maps induce subordination on quadratic forms.

Lifts and pushdowns preserve key properties of quadratic forms.

The theory is extended to tropical and supertropical algebra settings.

Abstract

The paper expands the theory of quadratic forms on modules over a semiring R, introduced in [12]-[14], especially in the setup of tropical and supertropical algebra. Isometric linear maps induce subordination on quadratic forms, and provide a main tool in our current study. These maps allow lifts and pushdowns of quadratic forms on different modules, preserving basic characteristic properties.