# Basic Operations on Supertropical Quadratic Forms

### Zur Izhakian

Institute of Mathematics, University of Aberdeen, AB24 3UE Aberdeen, UK### Manfred Knebusch

Department of Mathematics, NWF-I Mathematik, Universität Regensburg, 93040 Regensburg

## Abstract

In the case that a module $V$ over a (commutative) supertropical semiring $R$ is free, the $R$-module Quad($V$) of all quadratic forms on $V$ is almost never a free module. Nevertheless, Quad($V$) has two free submodules, the module QL($V$) of quasilinear forms with base $D_{0}$ and the module Rig($V$) of rigid forms with base $H_{0}$, such that Quad($V$)=QL($V$)+Rig($V$) and QL($V$)$∩$ Rig($V$) =${0}$. In this paper we study endomorphisms of Quad($V$) for which each submodule $Rq$ with $q∈D_{0}∪H_{0}$ is invariant; these basic endomorphisms are determined by coefficients in $R$ and do not depend on the base of $V$. We aim for a description of all basic endomorphisms of Quad($V$), or more generally of its submodules spanned by subsets of $D_{0}∪H_{0}$. But, due to complexity issues, this naive goal is highly nontrivial for an arbitrary supertropical semiring $R$. Our main stress is therefore on results valid under only mild conditions on $R$, while a complete solution is provided for the case that $R$ is a tangible supersemifield.

## Cite this article

Zur Izhakian, Manfred Knebusch, Basic Operations on Supertropical Quadratic Forms. Doc. Math. 22 (2017), pp. 1661–1707

DOI 10.4171/DM/607