### Abstract

In the case that a module $V$ over a (commutative) supertropical semiring $R$ is free, the $R$-module $\QV$ of all quadratic forms on $V$ is almost never a free module. Nevertheless, $\QV$ has two free submodules, the module $\QL(V)$ of quasilinear forms with base $\mfDz$ and the module $\Rig(V)$ of rigid forms with base $\mfHz$, such that $\QV = \QL(V) + \Rig(V)$ and $\QL(V) \cap \Rig(V) = \00.$ In this paper we study endomorphisms of $\QV$ for which each submodule $Rq$ with $q \in \mfDz \cup \mfHz$ 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 $\QV$, or more generally of its submodules spanned by subsets of $\mfDz \cup \mfHz$. 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.

Original language | English |
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Pages (from-to) | 1661-1707 |

Number of pages | 47 |

Journal | Documenta Mathematica |

Volume | 22 |

Publication status | Published - 2017 |

### Keywords

- Tropical algebra
- supertropical modules
- bilinear forms
- quadratic forms
- quadratic pairs
- minimal ordering
- unique base property

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## Cite this

Izhakian, Z., & Knebusch, M. (2017). Basic Operations on Supertropical Quadratic Forms.

*Documenta Mathematica*,*22*, 1661-1707. https://www.math.uni-bielefeld.de/documenta/vol-22/51.html