Basic Operations on Supertropical Quadratic Forms

Zur Izhakian, Manfred Knebusch

Research output: Contribution to journalArticlepeer-review


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
Publication statusPublished - 2017

Bibliographical 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.


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


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