Minimal orderings and quadratic forms on a free module over a supertropical semiring

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• 作者：Zur Izhakian^a ; ^{zzur@abdn.ac.uk" class="auth_mail" title="E-mail the corresponding author} ; ^{zzur@math.biu.ac.il" class="auth_mail" title="E-mail the corresponding author} ; Manfred Knebusch^b ; ^{manfred.knebusch@mathematik.uni-regensburg.de" class="auth_mail" title="E-mail the corresponding author} ; Louis Rowen^c ; ^{rowen@math.biu.ac.il" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 15A03 ; 15A09 ; 15A15 ; 16Y60 ; secondary ; 14T05 ; 15A33 ; 20M18 ; 51M20
• 刊名：Linear Algebra and its Applications
• 出版年：2016
• 出版时间：15 October 2016
• 年：2016
• 卷：507
• 期：Complete
• 页码：420-461
• 全文大小：665 K

文摘

This paper is a sequel to [6], in which we introduced quadratic forms on a module over a supertropical semiring R   and analyzed the set of bilinear companions of a single quadratic form V→R$V\to R$ in case the module V is free. Any (semi)module over a semiring gives rise to what we call its minimal ordering, which is a partial order iff the semiring is “upper bound.” Any polynomial map q (or quadratic form) then induces a pre-order, which can be studied in terms of “q-minimal elements,” which are elements a   which cannot be written in the form b+c$b+c$ where b<a$b<a$ but q(b)=q(a)$q(b)=q(a)$. We determine the q-minimal elements by examining their support.

But the class of all   polynomial maps (in up to rank(V)$\mathrm{rank}(V)$ variables) is itself a module over R, so the basic properties of the minimal ordering are applied to this R  -module, or its submodule Quad(V)$\mathrm{Quad}(V)$ consisting of quadratic forms on V. This is a significant initial step in the classification of quadratic forms over semirings arising in tropical mathematics.

Quad(V)$\mathrm{Quad}(V)$ is the sum of two disjoint submodules QL(V)$\mathrm{QL}(V)$ and Rig(V)$\mathrm{Rig}(V)$, consisting of the quasilinear and the rigid quadratic forms on V respectively (cf. [6]). Both QL(V)$\mathrm{QL}(V)$ and Rig(V)$\mathrm{Rig}(V)$ are free with explicitly known bases, but Quad(V)$\mathrm{Quad}(V)$ itself is almost never free.