Basic Operations on Supertropical Quadratic Forms

  • Zur Izhakian

    Institute of Mathematics, University of Aberdeen, AB24 3UE Aberdeen, UK
  • Manfred Knebusch

    Department of Mathematics, NWF-I Mathematik, Universität Regensburg, 93040 Regensburg
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In the case that a module over a (commutative) supertropical semiring is free, the -module Quad() of all quadratic forms on is almost never a free module. Nevertheless, Quad() has two free submodules, the module QL() of quasilinear forms with base and the module Rig() of rigid forms with base , such that Quad()=QL()+Rig() and QL() Rig() =. In this paper we study endomorphisms of Quad() for which each submodule with is invariant; these basic endomorphisms are determined by coefficients in and do not depend on the base of . We aim for a description of all basic endomorphisms of Quad(), or more generally of its submodules spanned by subsets of . But, due to complexity issues, this naive goal is highly nontrivial for an arbitrary supertropical semiring . Our main stress is therefore on results valid under only mild conditions on , while a complete solution is provided for the case that is a tangible supersemifield.

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Zur Izhakian, Manfred Knebusch, Basic Operations on Supertropical Quadratic Forms. Doc. Math. 22 (2017), pp. 1661–1707

DOI 10.4171/DM/607