Basic Operations on Supertropical Quadratic Forms

Zur Izhakian, Manfred Knebusch

Research output: Contribution to journalArticle

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 languageEnglish
Pages (from-to)1661-1707
Number of pages47
JournalDocumenta Mathematica
Volume22
Publication statusPublished - 2017

Fingerprint

Quadratic form
Endomorphisms
Module
Semiring
Valid
Subset
Invariant
Arbitrary
Coefficient
Form

Keywords

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

Cite this

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

In: Documenta Mathematica, Vol. 22, 2017, p. 1661-1707.

Research output: Contribution to journalArticle

Izhakian, Zur ; Knebusch, Manfred. / Basic Operations on Supertropical Quadratic Forms. In: Documenta Mathematica. 2017 ; Vol. 22. pp. 1661-1707.
@article{b03d66b3312743bd87e6a7959ea1b10e,
title = "Basic Operations on Supertropical Quadratic Forms",
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.",
keywords = "Tropical algebra, supertropical modules, bilinear forms, quadratic forms, quadratic pairs, minimal ordering, unique base property",
author = "Zur Izhakian and Manfred Knebusch",
note = "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.",
year = "2017",
language = "English",
volume = "22",
pages = "1661--1707",
journal = "Documenta Mathematica",
issn = "1431-0635",
publisher = "Deutsche Mathematiker Vereinigung",

}

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 -