Supertropical Quadratic Forms I

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 $\rho$: $R \to U$ with values in a supertropical semiring [5], we define - after choosing a base $\mathcal{L}=(v_i|i \in I)$ 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 $\mathcal{L}$ via ${\varphi}$.