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 VV over a (commutative) supertropical semiring RR is free, the RR-module Quad(VV) of all quadratic forms on VV is almost never a free module. Nevertheless, Quad(VV) has two free submodules, the module QL(VV) of quasilinear forms with base D0\frak{D}_0 and the module Rig(VV) of rigid forms with base H0\frak{H}_0, such that Quad(VV)=QL(VV)+Rig(VV) and QL(VV)\cap Rig(VV) ={0}\{0\}. In this paper we study endomorphisms of Quad(VV) for which each submodule RqRq with qD0H0q \in \frak{D}_0\cup\frak{H}_0 is invariant; these basic endomorphisms are determined by coefficients in RR and do not depend on the base of VV. We aim for a description of all basic endomorphisms of Quad(VV), or more generally of its submodules spanned by subsets of D0H0\frak{D}_0\cup\frak{H}_0. But, due to complexity issues, this naive goal is highly nontrivial for an arbitrary supertropical semiring RR. Our main stress is therefore on results valid under only mild conditions on RR, while a complete solution is provided for the case that RR 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