Abstract
| Original language | English |
|---|---|
| Pages (from-to) | 1661-1707 |
| Number of pages | 47 |
| Journal | Documenta Mathematica |
| Volume | 22 |
| Publication status | Published - 2017 |
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Keywords
- Tropical algebra
- supertropical modules
- bilinear forms
- quadratic forms
- quadratic pairs
- minimal ordering
- unique base property
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Basic Operations on Supertropical Quadratic Forms. / Izhakian, Zur; Knebusch, Manfred.
In: Documenta Mathematica, Vol. 22, 2017, p. 1661-1707.Research output: Contribution to journal › Article
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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 -