Supertropical quadratic forms I

详细信息    查看全文

• 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 ; 11D09 ; 11E81 ; 15A63 ; 15A03 ; 15A15 ; 16Y60 ; secondary ; 14T05 ; 13F30 ; 16W60
• 刊名：Journal of Pure and Applied Algebra
• 出版年：2016
• 出版时间：January 2016
• 年：2016
• 卷：220
• 期：1
• 页码：61-93
• 全文大小：619 K

文摘

We initiate the theory of a quadratic form q over a semiring R, with a view to study tropical linear algebra. As customary, one can write where b is a companion bilinear form. In contrast to the classical theory of quadratic forms over a field, the companion bilinear form need not be uniquely defined. Nevertheless, q   can always be written as a sum of quadratic forms q=q_QL+蟻$q=q_{\mathrm{QL}}+蟻$, where q_QL$q_{\mathrm{QL}}$ is quasilinear in the sense that q_QL(x+y)=q_QL(x)+q_QL(y)$q_{\mathrm{QL}}(x+y)=q_{\mathrm{QL}}(x)+q_{\mathrm{QL}}(y)$, and 蟻 is rigid in the sense that it has a unique companion. In case that R   is supertropical, we obtain an explicit classification of these decompositions q=q_QL+蟻$q=q_{\mathrm{QL}}+蟻$ and of all companions b of q, and see how this relates to the tropicalization procedure.