Geodesic
Basic Operations on Supertropical Quadratic Forms
Documenta mathematica, Tome 22 (2017), pp. 1661-1707
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In the case that a module V over a (commutative) supertropical semiring R is free, the R-module Quad(V) of all quadratic forms on V is almost never a free module. Nevertheless, Quad(V) has two free submodules, the module QL(V) of quasilinear forms with base D0​ and the module Rig(V) of rigid forms with base H0​, such that Quad(V)=QL(V)+Rig(V) and QL(V)∩ Rig(V) ={0}. In this paper we study endomorphisms of Quad(V) for which each submodule Rq with q∈D0​∪H0​ 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 Quad(V), or more generally of its submodules spanned by subsets of D0​∪H0​. 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.
DOI : 10.4171/dm/607
Classification : 15A63, 16Y60, 20M18
Mots-clés : quadratic forms, bilinear forms, tropical algebra, supertropical modules, quadratic pairs, minimal ordering, unique base property 
@article{10_4171_dm_607,
     author = {Zur Izhakian and Manfred Knebusch},
     title = {Basic {Operations} on {Supertropical} {Quadratic} {Forms}},
     journal = {Documenta mathematica},
     pages = {1661--1707},
     year = {2017},
     volume = {22},
     doi = {10.4171/dm/607},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/dm/607/}
}
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Zur Izhakian; Manfred Knebusch. Basic Operations on Supertropical Quadratic Forms. Documenta mathematica, Tome 22 (2017), pp. 1661-1707. doi: 10.4171/dm/607

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