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Supertropical Quadratic Forms I
Rings and Algebras · Submitted 23 Sep 2013
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Nevertheless, $q$ can always be written as a sum of quadratic forms $q = \kappa + \rho,$ where $\kappa$ is quasilinear in the sense that $\kappa(x+y) = \kappa(x) + \kappa(y),$ and $\rho$ is rigid in the sense that it has a unique companion.
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We initiate the theory of a quadratic form $q$ over a semiring $R$. As customary, one can write $$q(x+y) = q(x) + q(y)+ b(x,y),$$ where $b$ is a companion bilinear form. But in contrast to the ring-theoretic case, the companion bilinear form need not be uniquely defined. Nevertheless, $q$ can always be written as a sum of quadratic forms $q = \kappa + \rho,$ where $\kappa$ is quasilinear in the sense that $\kappa(x+y) = \kappa(x) + \kappa(y),$ and $\rho$ is rigid in the sense that it has a unique companion. In case that $R$ is a supersemifield (cf. Definition 4.1 below) and $q$ is defined on a free $R$-module, we obtain an explicit classification of these decompositions $q = \kappa + \rho $ and of all companions $b$ of $q$. As an application to tropical geometry, given a quadratic form $q: V \to R$ on a free module $V$ over a commutative ring $R$ and a supervaluation $\varphi:R \to U$ with values in a supertropical semiring [5], we define - after choosing a base $L=(v_i | i\in I)$ of $V$ - a quadratic form $q^\varphi: U^{(I)} \to U$ on the free module $U^{(I)}$ over the semiring $U$. The analysis of quadratic forms over a supertropical semiring enables one to measure the "position" of $q$ with respect to $L$ via $\varphi$.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We initiate the theory of a quadratic form $q$ over a semiring $R$. | contribution 0.99 | 0.94 | 0.05 | asserted 0.99 |
| s1 | As customary, one can write $$q(x+y) = q(x) + q(y)+ b(x,y),$$ where $b$ is a companion bilinear form. | background 1.00 | 0.10 | 0.03 | asserted 0.90 |
| s2 | But in contrast to the ring-theoretic case, the companion bilinear form need not be uniquely defined. | result 0.46 | 0.39 | 0.06 | asserted 0.76 |
| s3 | Nevertheless, $q$ can always be written as a sum of quadratic forms $q = \kappa + \rho,$ where $\kappa$ is quasilinear in the sense that $\kappa(x+y) = \kappa(x) + \kappa(y),$ and $\rho$ is rigid in the sense that it has a unique companion. | result 0.91 | 0.85 | 0.45 | asserted 0.99 |
| s4 | In case that $R$ is a supersemifield (cf. Definition 4.1 below) and $q$ is defined on a free $R$-module, we obtain an explicit classification of these decompositions $q = \kappa + \rho $ and of all companions $b$ of $q$. | result 0.81 | 0.93 | 0.30 | asserted 1.00 |
| s5 | As an application to tropical geometry, given a quadratic form $q: V \to R$ on a free module $V$ over a commutative ring $R$ and a supervaluation $\varphi:R \to U$ with values in a supertropical semiring [5], we define - after choosing a base $L=(v_i | i\in I)$ of $V$ - a quadratic form $q^\varphi: U^{(I)} \to U$ on the free module $U^{(I)}$ over the semiring $U$. | contribution 0.65 | 0.87 | 0.06 | asserted 1.00 |
| s6 | The analysis of quadratic forms over a supertropical semiring enables one to measure the "position" of $q$ with respect to $L$ via $\varphi$. | interpretation 0.86 | 0.70 | 0.13 | asserted 0.82 |
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The paper
Original abstract
We initiate the theory of a quadratic form $q$ over a semiring $R$. As customary, one can write $$q(x+y) = q(x) + q(y)+ b(x,y),$$ where $b$ is a companion bilinear form. But in contrast to the ring-theoretic case, the companion bilinear form need not be uniquely defined. Nevertheless, $q$ can always be written as a sum of quadratic forms $q = \kappa + \rho,$ where $\kappa$ is quasilinear in the sense that $\kappa(x+y) = \kappa(x) + \kappa(y),$ and $\rho$ is rigid in the sense that it has a unique companion. In case that $R$ is a supersemifield (cf. Definition 4.1 below) and $q$ is defined on a free $R$-module, we obtain an explicit classification of these decompositions $q = \kappa + \rho $ and of all companions $b$ of $q$. As an application to tropical geometry, given a quadratic form $q: V \to R$ on a free module $V$ over a commutative ring $R$ and a supervaluation $\varphi:R \to U$ with values in a supertropical semiring [5], we define - after choosing a base $L=(v_i | i\in I)$ of $V$ - a quadratic form $q^\varphi: U^{(I)} \to U$ on the free module $U^{(I)}$ over the semiring $U$. The analysis of quadratic forms over a supertropical semiring enables one to measure the "position" of $q$ with respect to $L$ via $\varphi$.
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How the literature uses it
2 arXiv papers cite this in 33 sentences
Papers citing it · 26 sentences without a confident label
Builds on it 1 paper
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It is a sequel to the paper [ref] by the same authors.
Supertropical Quadratic Forms II · Introduction · 0.82
Cites it as support 1 paper
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* If $q$ is diagonally zero, then $q$ is rigid, cf. [ref] .
Quadratic and Symmetric Bilinear Forms on Modules with Unique Base Over a Semiring · Indecomposability in tensor products · 0.96
Cites it as background 2 papers
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We recall from [ref] that, when $V$ is free with base $(_i : i ∈I)$, then $q$ admits a (not necessarily unique) balanced companion, i.e., a companion $b : V ×V →R$ such that $b(x,x) = 2q(x)$ for all $x ∈V$, and that it suffices to know for this that $b(_i,_i) = 2q( _i)$ for all $i ∈I$ [ref] .
Quadratic and Symmetric Bilinear Forms on Modules with Unique Base Over a Semiring · Expansions and tensor products · 0.85
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In the paper [ref] , the main thrust is the study of the set of all companions of a given quadratic form $q$ on a free module $V$ over a supertropical semiring $R.$ After fixing a base $(ε_ii∈I)$ of $V$, this set can be described by use of a “companion matrix” ($C_i,j(q))$, cf. 6.
Supertropical Quadratic Forms II · Introduction · 0.84
What it cites · 4 sentences without a confident label
Cites as background 1 paper
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Supertropical semirings allow a refinement of valuation theory to a theory of “supervaluations”, the basics of which can be found in [ref] – [ref] .
Monoid Valuations and Value Ordered Supervaluations · Introduction and basic notions · 0.97
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