### Abstract

Original language | English |
---|---|

Pages (from-to) | 1661-1707 |

Number of pages | 47 |

Journal | Documenta Mathematica |

Volume | 22 |

Publication status | Published - 2017 |

### Fingerprint

### Keywords

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

### Cite this

*Documenta Mathematica*,

*22*, 1661-1707.

**Basic Operations on Supertropical Quadratic Forms.** / Izhakian, Zur; Knebusch, Manfred.

Research output: Contribution to journal › Article

*Documenta Mathematica*, vol. 22, pp. 1661-1707.

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TY - JOUR

T1 - Basic Operations on Supertropical Quadratic Forms

AU - Izhakian, Zur

AU - Knebusch, Manfred

N1 - The research of the first author has been supported by the Research Councils UK (EPSRC), grant no EP/N02995X/1. The authors thank the referee for many helpful suggestions.

PY - 2017

Y1 - 2017

N2 - 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.

AB - 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.

KW - Tropical algebra

KW - supertropical modules

KW - bilinear forms

KW - quadratic forms

KW - quadratic pairs

KW - minimal ordering

KW - unique base property

M3 - Article

VL - 22

SP - 1661

EP - 1707

JO - Documenta Mathematica

JF - Documenta Mathematica

SN - 1431-0635

ER -